nobreakspace

# On the Resolution Graph of a Plane Curve

João Cabral and Orlando Neto and Pedro C. Silva<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We show that the resolution graph of a plane curve singularity admits a
canonical decomposition into elementary graphs.

##  1\. Introduction

The main purpose of this paper is to give a canonical decomposition of the
resolution graph of an arbitrary plane curve that generalizes the
decomposition of the resolution graph of an irreducible plane curve introduced
by Brieskorn and Knorrer remark in [1]. The elementary graphs of this
decomposition are of two types: smooth type (all branches are smooth, see
Example 4.4) and cusp type (all branches are ”cusps” and the graph is linear,
see Example 6.6).

One can find in [4] a closed form description for the local fundamental group
of a plane curve, given the Puiseux expansions of the branches of the curve or
its Eggers tree. One of the motivations of this paper is to clear the way to
give a closed form description of the local fundamental group of a plane curve
from its equations (or its resolution graph) that can be implemented in a
computer ([2]). Yang Jingen gave in [6] an interesting characterization of the
resolution graphs.

Section 2 recalls several topological invariants of plane curves and its
behaviour by blow up. Theorem 3.6 shows that a normal crossings graph is a
resolution graph if and only if it is contractible. Readers willing to accept
this result may skip most of sections 2 and 3. Sections 4, 5 and 6 are
dedicated to the characterization of the elementary graphs. Section 7 proves
the main result.

##  2\. Invariants of Plane Curve Singularities

Let $Y$ be the germ of an irreducible plane curve defined by
$f\in\mathbb{C}\\{x,y\\}$. Assume that the tangent cone of $Y$ is transversal
to $\\{x=0\\}$ and $Y$ has multiplicity $k$. By the Puiseux Theorem there are
a positive integer $n$ and complex numbers $c_{i}$, $i\geq n$, such that
$(n,k)=1$, $c_{n}\not=0$ and $Y$ admits the local parametrization

(1) $x=t^{k},\qquad y=\sum_{i\geq n}c_{i}t^{i}.$

Consider the injective morphism from $\mathbb{C}\\{x\\}$ to
$\mathbb{C}\\{t\\}$ that takes $x$ into $t^{k}$. We will sometimes denote $t$
by $x^{\frac{1}{k}}$. For $\alpha\in\mathbb{Q}_{+}$, set $a_{\alpha}=c_{i}$ if
$\alpha=i/k$ and $a_{\alpha}=0$ if $k{\alpha}\not\in\mathbb{Z}$. We say that
(1) and $y=\sum_{\alpha}a_{\alpha}x^{\alpha}$ are Puiseux expansions of $Y$
relatively to the system of local coordinates $(x,y)$.

There is a certain ambiguity on the determination of the coefficients of the
Puiseux expansion of a plane curve. When we compare two Puiseux expansions

(2) $y=\sum_{\alpha}a_{\alpha}x^{\alpha}~{}\mbox{ and
}~{}y=\sum_{\alpha}b_{\alpha}x^{\alpha},$

we shall assume that $a_{\alpha}=b_{\alpha}$ for ${\alpha}\leq{\beta}$ if
$\sum_{{\alpha}\leq{\beta}}a_{\alpha}x^{\alpha}$ and
$\sum_{{\alpha}\leq{\beta}}b_{\alpha}x^{\alpha}$ are parametrizations of the
same plane curve (see Section 4.1 of [5]).

If $Y$ admits the Puiseux expansion $y=\sum_{\alpha}a_{{\alpha}}x^{\alpha}$,
there are unique rational numbers $\alpha_{1}<\cdots<\alpha_{g}$ such that:

  1. (1)

For $1\leq k\leq g$, $a_{{\alpha}_{k}}\not=0$.

  2. (2)

Set ${\alpha}_{g+1}=+\infty$. If $a_{\alpha}\not=0$ and
${\alpha}<{\alpha}_{k+1}$, ${\alpha}$ is a linear combination with integer
coefficients of ${\alpha}_{1},\ldots,{\alpha}_{k}$.

  3. (3)

$\alpha_{i}$, $i\in\\{2,\ldots,g\\}$, is not a linear combination with integer
coefficients of $\alpha_{1},\ldots,\alpha_{i-1}$.

The rational numbers $\alpha_{1},\ldots,\alpha_{g}$ are called the Puiseux
exponents of $Y$.

Given two plane curves $Y,Z$ with Puiseux expansions (2), we call contact
exponent of $Y$ and $Z$ to the smallest $\gamma$ such that
$a_{\gamma}\not=b_{\gamma}$. We will denote it by $o(Y,Z)$.

Let $Y,Z$ be two plane curves with Puiseux expansions (2). Given a positive
rational $\gamma$, $o(Y,Z)=\gamma$ if and only if
$y=\sum_{{\alpha}\leq\beta}a_{\alpha}x^{\alpha}$ and
$y=\sum_{{\alpha}\leq\beta}b_{\alpha}x^{\alpha}$ define the same curve for all
positive rational $\beta<\gamma$ but
$y=\sum_{{\alpha}\leq\gamma}a_{\alpha}x^{\alpha}$ and
$y=\sum_{{\alpha}\leq\gamma}b_{\alpha}x^{\alpha}$ define different curves.

Notice that $o(Y,Z)=1$ if and only if $Y$ and $Z$ have different tangent
cones.

######  Theorem 2.1.

Let $\widetilde{Y}$ be the strict transform of an irreducible plane curve $Y$.
Let $E$ be the exceptional divisor of the blow up. Assume that $Y$ has Puiseux
exponents ${\alpha}_{1},\ldots,{\alpha}_{g}$.

  1. (1)

If ${\alpha}_{1}\geq 2$, $\widetilde{Y}$ is transversal to the exceptional
divisor and has Puiseux exponents
${\alpha}_{1}-1,{\alpha}_{2}-1,\ldots,{\alpha}_{g}-1$.

  2. (2)

Assume that ${\alpha}_{1}<2$. Then $\widetilde{Y}$ is tangent to the
exceptional divisor and $o(\widetilde{Y},E)=1/({\alpha}_{1}-1)$.

    1. (a)

If $1/({\alpha}_{1}-1)$ is not an integer, the Puiseux exponents of
$\widetilde{Y}$ are

$1/({\alpha}_{1}-1),{\alpha}_{2}/({\alpha}_{1}-1)-1,\ldots,{\alpha}_{g}/({\alpha}_{1}-1)-1$.

    2. (b)

If $1/({\alpha}_{1}-1)$ is an integer, the Puiseuxs exponents of
$\widetilde{Y}$ are
${\alpha}_{2}/({\alpha}_{1}-1)-1,\ldots,{\alpha}_{g}/({\alpha}_{1}-1)-1$.

###### Proof.

It follows from Theorem 3.5.5 of [5]. ∎

######  Corollary 2.2.

Let $X$ be a smooth complex surface. Let $E$ be an exceptional divisor of $X$.
Let $\pi:X\to\underline{X}$ be the blow down of $X$ along $E$. Let $Y$ be the
germ of a irreducible plane curve at a point $\sigma$ of $E$. Assume that $Y$
has Puiseux exponents ${\alpha}_{1},\ldots,{\alpha}_{g}$.

  1. (1)

If $Y$ is transversal to $E$, $\pi(Y)$ has Puiseux exponents
${\alpha}_{1}+1,{\alpha}_{2}+1\ldots,{\alpha}_{g}+1$.

  2. (2)

Assume that $Y$ is tangent to $E$.

    1. (a)

If $o(Y,E)$ is not an integer, $o(Y,E)={\alpha}_{1}$ and $\pi(Y)$ has Puiseux
exponents
$1+1/{\alpha}_{1},({\alpha}_{2}+1)/{\alpha}_{1},\ldots,({\alpha}_{g}+1)/{\alpha}_{1}$.

    2. (b)

If $o(Y,E)$ is an integer, $\pi(Y)$ has Puiseux exponents

$1+1/o(Y,E),({\alpha}_{1}+1)/o(Y,E),\ldots,({\alpha}_{g}+1)/o(Y,E)$.

######  Theorem 2.3.

Let $Y,Z$ be irreducible plane curves with the same tangent cone. Let
$\widetilde{Y},\widetilde{Z}$ be their strict transforms and $E$ the
exceptional divisor. Let $\mu$ $[\nu,\widetilde{\mu},\widetilde{\nu}]$ be the
first Puiseux exponent of $Y$ $[Z,\widetilde{Y},\widetilde{Z}]$. Assume that
$\nu\leq\mu$.

  1. (1)

If $\nu\geq 2$, $o(\widetilde{Y},\widetilde{Z})=o({Y},{Z})-1$ and
$\widetilde{Y},\widetilde{Z}$ are transversal to $E$.

  2. (2)

If $\nu<2$ and $\mu>2$, $o(\widetilde{Y},\widetilde{Z})=1$ and $\widetilde{Z}$
is tangent to $E$.

  3. (3)

Assume that $\mu<2$. Then $\widetilde{Y},\widetilde{Z}$ are tangent to $E$.
Moreover, $\widetilde{\mu}\leq\widetilde{\nu}$.

    1. (a)

Assume that $o(Y,Z)>\mu$. Then $\mu=\nu$, $\widetilde{\mu}=\widetilde{\nu}$
and

$o(\widetilde{Y},E)=o(\widetilde{Z},E)=\widetilde{\mu}<o({Y},{Z})/(\mu-1)-1=o(\widetilde{Y},\widetilde{Z})$.

    2. (b)

Assume that $o(Y,Z)=\mu$. Then $\mu=\nu$ and

$o(\widetilde{Y},E)=o(\widetilde{Z},E)=\widetilde{\mu}=\widetilde{\nu}=o(\widetilde{Y},\widetilde{Z})$.

    3. (c)

Assume that $o(Y,Z)<\mu$. Then $o(Y,Z)=\nu$ and

$o(\widetilde{Y},E)=o(\widetilde{Y},\widetilde{Z})=\widetilde{\mu}<\widetilde{\nu}=o(\widetilde{Z},E)$.

###### Proof.

Statements $(1)$ and $(2)$ of Theorem 2.1 imply statement $(2)$. Let (1) be a
parametrization of $Y$. Therefore the curve $\widetilde{Y}$ has a
parametrization given by

(3) $x=t^{k},\qquad\frac{y}{x}=\sum_{i\geq n}c_{i}t^{i-k}=t^{n-k}\gamma(t),\
\gamma(0)\neq 0.$

Assume $\nu\geq 2$. Then the parametrization (3) is a Puiseux expansion and
statement $(1)$ holds. Assume $\mu<2$. In this case, the parametrization (3)
is not a Puiseux expansion. Let $x=t^{k},\
y=t^{n}\overline{\gamma}(t),\overline{\gamma}(0)\neq 0$ be a Puiseux expansion
of $Z$. Let $\alpha(t)=\sum_{i\geq
0}\alpha_{i}t^{i},\overline{\alpha}(t)=\sum_{i\geq
0}\overline{\alpha}_{i}t^{i}$ such that
$\alpha(t)^{n-k}=\gamma(t),\overline{\alpha}(t)^{n-k}=\overline{\gamma}(t)$.
Let $\beta(t)=\sum_{i\geq 0}\beta_{i}t^{i},\overline{\beta}(t)=\sum_{i\geq
0}\overline{\beta}_{i}t^{i}$, such that $t=w\beta(w)$ is a solution of
$w=t\alpha(t)$ and $t=w\overline{\beta}(w)$ is a solution of
$w=t\overline{\alpha}(t)$. To prove statement $(3)$ it suffices to show that
if $\alpha(t)\equiv\overline{\alpha}(t)$ mod $(t)^{i}$, $i\in\mathbb{N}$,
$\beta(t)\equiv\overline{\beta}(t)$ mod $(t)^{i}$. Also, if
$\alpha_{i+1}\neq\overline{\alpha}_{i+1}$,
$\beta_{i+1}\neq\overline{\beta}_{i+1}$.

We can check easily that $\alpha_{i}$ depends only on $c_{n},\ldots,c_{n+i}$,
for all $i\geq 0$, and $\alpha(0)\neq 0$. From the equality $w=t\alpha(t)$ we
get

(4) $w=\sum_{i\geq
0}\sum_{j_{0},\ldots,j_{i}}\alpha_{i}\beta_{j_{0}}\ldots\beta_{j_{i}}w^{j_{0}+\cdots+j_{i}+i+1}.$

From equality (4) we deduce the equations

(5) $\alpha_{0}\beta_{0}=1,$ (6)
$\alpha_{0}\beta_{k-1}+\sum_{i=1}^{k-1}\sum_{j_{0}+\cdots+j_{i}=k-1-i}\alpha_{i}\beta_{j_{0}}\ldots\beta_{j_{i}}=0,\
k\in\mathbb{N}.$

Therefore $\beta(0)\neq 0$ and the parametrization

$x=(w\beta(w))^{\frac{k}{(k,n-k)}},\;\textstyle{\frac{y}{x}}=w^{\frac{n-k}{(k,n-k)}}$

is a Puiseux expansion. Also, by induction, we prove that $\beta_{i}$ depends
only on $\alpha_{0},\ldots,\alpha_{i}$, for all $i\geq 0$. ∎

######  Corollary 2.4.

Let $X$ be a smooth complex surface. Let $E$ be an exceptional divisor of $X$.
Let $\pi:X\to\underline{X}$ be the blow down of $X$ along $E$. Let $Y,Z$ be
two germs of irreducible plane curves at a point $\sigma$ of $E$. The curves
$\pi(Y),\pi(Z)$ are germs of irreducible plane curves at the point $\sigma(E)$
of $\underline{X}$, with the same tangent cone. Let $\mu$ $[\nu]$ be the first
Puiseux exponent of $Y$ $[Z]$. Assume that $\mu\leq\nu$.

  1. (1)

If $Y,Z$ are transversal to $E$, $o(\pi(Y),\pi(Z))=o(Y,Z)+1$.

  2. (2)

If $Y$ is transversal to $E$ and $Z$ is tangent to $E$,
$o(\pi(Y),\pi(Z))=1+1/\nu$.

  3. (3)

Assume that $Y$ and $Z$ are tangent to $E$.

    1. (a)

If $o(Y,E)=o(Z,E)<o(Y,Z)$, $o(\pi(Y),\pi(Z))=(\mu-1)(o(Y,Z)+1)$.

    2. (b)

If $o(Y,E)=o(Z,E)=o(Y,Z)$, $o(\pi(Y),\pi(Z))=1+1/o(Y,Z)$.

    3. (c)

If $o(Y,E)=o(Y,Z)<o(Z,E)$, $o(\pi(Y),\pi(Z))=1+1/o(Y,Z)$.

##  3\. The Combinatorics of the Resolution Algorithm

######  3.1.

Let $Y$ be the germ at the origin of a plane curve defined by
$f\in\mathbb{C}\\{x,y\\}$. Let $X$ be a polydisc where $f$ converges. We can
assume that $Y$ has no singular points on $X\setminus\\{(0,0)\\}$. Set
$X_{0}=X$, $Y_{0}=Y$, ${\sigma}_{0}=(0,0)$. Let $\pi_{1}:X_{1}\to X_{0}$ be
the blow up of $X$ with center ${\sigma}_{0}$. Set
$D_{1}=\pi_{1}^{-1}({\sigma}_{0})$. Let $Y_{1}$ be the strict transform of
$Y$.

Let us define recursively a sequence of triples
$(X_{\ell},D_{\ell},Y^{(\ell)})$, where $X_{\ell}$ is a smooth complex
surface, $D_{\ell}$ is a divisor with normal crossings of $X_{\ell}$ and
$Y^{(\ell)}$ is a union of germs of plane curves at points of $D_{\ell}$. Let
$W_{\ell}$ be set of points of $D_{\ell}\cap Y_{\ell}$ where $D_{\ell}\cup
Y^{(\ell)}$ is not a divisor with normal crossings. Let
$\pi_{\ell+1}:X_{\ell+1}\to X_{\ell}$ be the blow up of $X_{\ell}$ with center
$C_{\ell}$. Let $Y^{(\ell+1)}$ be the strict transform of $Y^{(\ell)}$ by
$\pi_{\ell+1}$. Set $W_{\ell+1}=\pi^{-1}(D_{\ell})$. After a finite number of
steps $D_{\ell}\cup Y^{(\ell)}$ is a normal crossings divisor, $W_{\ell}$ is
the empty set and $X_{\ell+1}=X_{\ell}$.

Let us describe this procedure in purely combinatorial terms.

######  Definition 3.1.

Consider the following data:

  1. (1)

A weighted tree $\mathcal{D}$. The vertices of $\mathcal{D}$ are called
divisors. The weight ${\omega}_{E}$ of a divisor $E$ of $\mathcal{D}$ is a
negative integer. We call ${\omega}_{E}$ the self-intersection number of $E$.
If two divisors are connected by an edge, we say that they intersect each
other.

  2. (2)

A nonempty set $o_{\mathcal{D}}$. The elements of $o_{\mathcal{D}}$ are called
points of $\mathcal{D}$. If $\mathcal{D}$ is the empty graph,
$o_{\mathcal{D}}$ has only one element. We associate to each point $\sigma$ of
$\mathcal{D}$ a set of divisors $\mathcal{D}_{\sigma}$ of $\mathcal{D}$. If
$E\in\mathcal{D}_{\sigma}$, we say that $\sigma$ belongs to $E$ and $E$
contains $\sigma$.

  3. (3)

A family of disjoint finite sets $\mathcal{C}_{\sigma}$, where $\sigma$ is a
point of $\mathcal{D}$. The elements of $\mathcal{C}_{\sigma}$ are called
branches. We associate to each branch $i$ of $\mathcal{C}_{\sigma}$ a
nonnegative integer $g_{i}$ and rational numbers ${\varepsilon}_{i,j}$, $1\leq
j\leq g_{i}$. We call the rational numbers ${\varepsilon}_{i,j}$ the Puiseux
exponents of the branch $i$. Let $\widetilde{\mathcal{C}}_{\sigma}$ be the
union of $\mathcal{C}_{\sigma}$ with the set of divisors of $\mathcal{D}$ that
contain $\sigma$. We define a map
$o:\\{(i,j):i,j\in\widetilde{\mathcal{C}}_{\sigma},\>i\neq
j\\}\rightarrow\mathbb{Q}$. We call $o(i,j)$ the contact order of $i$ and $j$.

We call the set of data (1),(2),(3) a resolution step if it verifies the
conditions:

  1. (1)

A point of $\mathcal{D}$ belongs to at most two divisors. Given two divisors,
there is at most one point that belongs to both.

  2. (2)

If a point of $\mathcal{D}$ belongs to two divisors, they intersect each
other.

  3. (3)

If the graph $\mathcal{D}$ is nonempty, each point of $\mathcal{D}$ belongs to
some divisor.

  4. (4)

$o(i,j)\geq 1$ for each $i,j\in\widetilde{\mathcal{C}}_{\sigma}$.

  5. (5)

$o(i,j)=o(j,i)$ for each $i,j\in\widetilde{\mathcal{C}}_{\sigma}$.

  6. (6)

$o(i,k)\geq$min$\\{o(i,j),o(j,k)\\}$ for each
$i,j,k\in\widetilde{\mathcal{C}}_{\sigma}$.

  7. (7)

$1={\varepsilon}_{i,0}<{\varepsilon}_{i,1}<\cdots<{\varepsilon}_{i,g_{i}}$ for
each $i\in\mathcal{C}_{\sigma}$.

  8. (8)

${\varepsilon}_{i,j}$ is not a linear combination with integer coefficients of
${\varepsilon}_{i,1},\ldots,{\varepsilon}_{i,j-1}$ for $1\leq j\leq g_{i}$.

We say that a resolution step is a plane curve if the graph $\mathcal{D}$ is
the empty graph. We say that a resolution step has normal crossings at a point
$\sigma$ of $\mathcal{D}$ if the point $\sigma$ belongs to exactly one divisor
$E$, the point $\sigma$ has exactly one branch $Y$and $o(Y,E)=1$. We say that
a resolution step is a normal crossings graph if it is normal crossings at
each point $\sigma$ of $o_{\mathcal{D}}$. We will represent a normal crossings
graph in the following way. We will connect by an edge the asterisc that
represents a branch of $Y$ to the vertex that represents a divisor if there is
a point that belongs to both.

Let $\sigma$ be a point of $\mathcal{D}$. Consider in $\mathcal{C}_{\sigma}$
the equivalence relation $i\sim j$ if $o(i,j)>1$. We call an equivalence class
$\ell$ of $\mathcal{C}_{\sigma}$ a tangent line of $\sigma$.

Let $X$ be a smooth complex surface. Let $D$ be a normal crossings divisor of
$X$. We will call divisors of $D$ to the irreducible components of $D$. Let
$Y$ be the union of a finite set of germs of plane curves of $X$ at points of
$D$. If $D$ is not empty, $Y=\cup_{{\sigma}\in A}Y_{\sigma}$, where $A$ is a
finite subset of $D$ and $Y_{\sigma}$ is the germ at $\sigma$ of a plane curve
with no irreducible component contained in $D$. If $D$ is empty, $Y$ is the
germ of a plane curve at a point ${\sigma}_{0}$ of $X$.

Let us associate to a triple $(X,D,Y)$ a resolution step denoted by $[X,D,Y]$.
Let $\mathcal{D}$ be the graph with vertices the divisors of $D$. Two vertices
of $\mathcal{D}$ are connected if they intersect each other. We will label
each divisor $E$ of $D$ with its self-intersection number ${\omega}_{E}$ (see
[7]). If $D$ is not empty, set $o_{\mathcal{D}}=A$. Otherwise set
$o_{\mathcal{D}}=\\{{\sigma}_{0}\\}$. Let $\mathcal{D}_{\sigma}$ be the set of
divisors of $D$ that contain ${\sigma}$, for each ${\sigma}\in
o_{\mathcal{D}}$. Let $Y_{i}$, $i\in\mathcal{C}_{\sigma}$, be the irreducible
components of $Y_{\sigma}$. Let ${\varepsilon}_{ij}$, $1\leq j\leq g_{i}$, be
the Puiseux exponents of $Y_{i}$ for each $i\in C_{\sigma}$. If
$i,j\in\mathcal{C}_{\sigma}$ and $i\not=j$, set $o(i,j)=o(Y_{i},Y_{j})$. If
$i\in\mathcal{C}_{\sigma}$ and $E\in\mathcal{D}_{\sigma}$, set
$o(i,E)=o(Y_{i},E)$. If $E,F\in\mathcal{D}_{\sigma}$ and $E\not=F$, set
$o(E,F)=1$.

Notice that $[X,D,Y]$ is normal crossings if and only if $D\cup Y$ is a normal
crossings divisor. Moreover, $[X,D,Y]$ is a first step if and only if $D$ is
empty.

######  Definition 3.2.

Let us define the blow up
$(\overline{\mathcal{D}},(\overline{\mathcal{C}}_{\sigma}))$ of a resolution
step $(\mathcal{D},(\mathcal{C}_{\sigma}))$. Let $c_{\mathcal{D}}$ be the set
of ${\sigma}\in o_{\mathcal{D}}$ such that
$(\mathcal{D},(\mathcal{C}_{\sigma}))$ is not normal crossings at ${\sigma}$.
The vertices of $\overline{\mathcal{D}}$ will be the vertices of $\mathcal{D}$
and the elements of $c_{\mathcal{D}}$. For each divisor $E$ of
$\overline{\mathcal{D}}\setminus c_{\mathcal{D}}$ we set
$\overline{{\omega}}_{E}={\omega}_{E}-k$, where $k$ is the number of elements
of $c_{\mathcal{D}}$ that belong to $E$. We set $\overline{{\omega}}_{E}=-1$
for each element $E$ of $c_{\mathcal{D}}$. Let ${\sigma}\in c_{\mathcal{D}}$.
If ${\sigma}$ belongs to a divisor $E$, we connect ${\sigma}$ to $E$ by an
edge. If ${\sigma}$ belongs to a pair of divisors $E,F$, we withdraw the edge
that connects $E$ to $F$ and set

$o_{\overline{\mathcal{D}}}=(o_{\mathcal{D}}\setminus
c_{\mathcal{D}})\cup\cup_{{\sigma}\in c_{\mathcal{D}}}\\{\mbox{ tangent lines
of }{\sigma}\\}.$

We set $\overline{\mathcal{C}}_{\sigma}=\mathcal{C}_{\sigma}$ for each
${\sigma}\in o_{\mathcal{D}}\setminus c_{\mathcal{D}}.$ We set
$\overline{\mathcal{C}}_{\ell}=\ell$ for each tangent line $\ell$ of
${\sigma}\in c_{\mathcal{D}}$. We define the Puiseux exponents and the contact
exponents of $(\overline{\mathcal{D}},(\overline{\mathcal{C}}))$ according to
Theorems 2.1 and 2.3.

######  Definition 3.3.

Let us define the blow down
$(\underline{\mathcal{D}},(\underline{\mathcal{C}}_{\sigma}))$ of the
resolution step $(\mathcal{D},(\mathcal{C}_{\sigma}))$. The vertices of
$\underline{\mathcal{D}}$ are the vertices $E$ of $\mathcal{D}$ such that
${\omega}_{E}\not=-1$. For each vertex $E$ of $\underline{\mathcal{D}}$ we set
$\underline{{\omega}}_{E}={\omega}_{E}+k$, where $k$ is the number of divisors
$F\in{\mathcal{D}}\setminus\underline{\mathcal{D}}$ such that $F$ intersects
$E$. If two divisors $E,F$ of $\underline{\mathcal{D}}$ intersect a divisor of
weight $-1$ of $\mathcal{D}$, we connect $E$ and $F$ by an edge. Let
$o^{*}_{\mathcal{D}}$ be the set of points ${\sigma}\in o_{\mathcal{D}}$ such
that ${\sigma}$ does not belong to a divisor of weight $-1$. We set
$o_{\underline{\mathcal{D}}}=o^{*}_{\mathcal{D}}\cup({\mathcal{D}}\setminus\underline{\mathcal{D}})$.
If ${\sigma}\in o^{*}_{\mathcal{D}}$, set
$\underline{\mathcal{C}}_{\sigma}={\mathcal{C}}_{\sigma}$. If
$E\in{\mathcal{D}}\setminus\underline{\mathcal{D}}$, i.e. $\omega_{E}=-1$, set
$\underline{\mathcal{C}}_{E}=\cup\\{{\mathcal{C}}_{\sigma}$: ${\sigma}$
belongs to $E\\}$. We define the Puiseux exponents and the contact exponents
of $(\underline{\mathcal{D}},(\underline{\mathcal{C}}_{\sigma}))$ according to
Corollaries 2.2 and 2.4.

The following result is an immediate consequence of the previous definitions.

######  Lemma 3.4.

The blow up of the resolution step of $(X,D,Y)$ equals the resolution step of
the blow up of $(X,D,Y)$. The blow down of the resolution step of $(X,D,Y)$
equals the resolution step of the blow down of $(X,D,Y)$.

######  Definition 3.5.

We say that a normal crossings graph is a resolution graph if there is a plane
curve that is transformed into the normal crossings graph by a sequence of
blow ups. We say that a normal crossings graph is contractible if there is a
sequence of blow downs that takes the normal crossings graph into a plane
curve. On the step previous to the last blow down, we get the empty graph and
the normal crossings graph reduces to a single vertex. We call this vertex the
root of the normal crossings graph.

The root of a resolution graph is therefore the divisor created by the first
blow up. We call root of a resolution step to the strict transform of the
divisor created by the first blow up.

######  Theorem 3.6.

A normal crossings graph is a resolution graph if and only if it is
contractible.

###### Proof.

If the normal crossings graph is contractible, we can blow it down into a
plane curve and reverse the procedure, obtaining the initial resolution graph.
∎

A resolution graph $\mathcal{R}$ is minimal if there is no divisor $E$ of
weigth $-1$ of $\mathcal{R}$ such that after blowing down $E$ we obtain a
normal crossings graph. Given the germ of a plane curve $Y$, there is one and
only one minimal resolution graph $\mathcal{R}_{Y}$ that is a resolution graph
of $Y$.

Given a divisor $E$ of a resolution step we call valence of $E$ to the number
of divisors of that resolution step that intersect $E$. We say that a
resolution step is linear if all of its divisors have valence $0,1$ or $2$. If
a resolution step has only one divisor, this divisor is simultaneously the
root and a terminal vertex of the resolution step. Otherwise we call terminal
vertex of a resolution step to its vertices of valence $1$ that are not the
root.

##  4\. Resolution Graphs of Smooth Type

Let $Y$ $[\Sigma]$ be a [smooth] plane curve of $(X,o)$. We call the pair
$(Y,\Sigma)$ a logarithmic plane curve if $Y$ and $\Sigma$ have no common
irreducible components. We say that a logarithmic plane curve is singular if
the curve $Y\cup\Sigma$ is not a normal crossings divisor. We call branches of
$(Y,\Sigma)$ to the branches of $Y$. Let $C$ be a smooth curve transversal to
$\Sigma$. Define $\tau_{C}(Y)$ as the smallest positive integer $\tau$ such
that $o(Y_{i},C)<\tau$ for each singular branch $Y_{i}$ of $Y$.

Let $\pi:\widetilde{X}\rightarrow X$ be the composition of blow ups that
desingularizes $Y\cup\Sigma$. Let $\widetilde{Y}$ [$\widetilde{\Sigma}$] be
the strict transform of $Y$ [$\Sigma$] by $\pi$. We call resolution graph of
$(Y,\Sigma)$ to the graph $\mathcal{R}^{\Sigma}_{Y}$ obtained in the following
manner: the vertices of $\mathcal{R}^{\Sigma}_{Y}$ are the vertices of
$\mathcal{R}_{Y\cup\Sigma}$ plus $\widetilde{\Sigma}$; the branches of
$\widetilde{Y}$ are the branches of $\widetilde{Y\cup\Sigma}$ minus
$\widetilde{\Sigma}$. We call $\widetilde{\Sigma}$ the root of
$\mathcal{R}^{\Sigma}_{Y}$. We set ${\omega}_{\widetilde{\Sigma}}=-c-1$, where
$c$ is the number of times we blow up a point of a strict transform of
$\Sigma$.

We call log resolution graph to a pair
$\mathcal{R}^{\Sigma}=(\mathcal{R},\Sigma)$, where:

$(i)$ $\mathcal{R}$ is a normal crossings graph;

$(ii)$ $\Sigma$ is a divisor of valence $1$ of $\mathcal{R}$;

$(iii)$ If we consider $\Sigma$ a branch of $\mathcal{R}$, $\mathcal{R}$
becomes the resolution graph of a plane curve;

$(iv)$ The image $\underline{\Sigma}$ of $\Sigma$, by the blow down of
$\mathcal{R}$ into a plane curve, is a smooth component of $Y$;

$(v)$ ${\omega}_{\Sigma}=-c-1$, where $c$ is the number of times necessary to
blow down a divisor intersecting $\Sigma$ when we blow down $\mathcal{R}$ into
a plane curve.

A normal crossings graph $\mathcal{R}$ is a log resolution graph if and only
if there is a logarithmic plane curve $(Y,\Sigma)$ such that
$\mathcal{R}=\mathcal{R}^{\Sigma}_{Y}$.

Let $\mathcal{D}$ be a resolution step and $E$ a smooth divisor of
$\mathcal{D}$ with weight $\omega_{E}$ and valence $\vartheta_{E}$. We call
rectified weight of E to the number $\omega_{E}+\vartheta_{E}$.

######  Definition 4.1.

Let $\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] be a [log] resolution graph. We say
that $\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] is a [log]elementary graph of
smooth type if the root of $\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] has
rectified weigth $-1$ and the other divisors of $\mathcal{R}$
[$\mathcal{R}^{\Sigma}$] have rectified weigth $0$.

######  Theorem 4.2.

Let $(Y,\Sigma)$ be a logarithmic plane curve. Assume $\Sigma$ smooth. The
graph $\mathcal{R}^{\Sigma}_{Y}$ is a log elementary graph of smooth type if
and only if all branches of $Y$ are non singular and transversal to $\Sigma$.

###### Proof.

We will use the notations of Paragraph 3.1. Set $\Sigma^{(0)}=\Sigma$. Let
$\Sigma^{(\ell+1)}$ be the strict transform of $\Sigma^{(\ell)}$ by
$\pi_{\ell+1}$, for each $\ell\geq 0$. Assume that all branches of $Y$ are non
singular and transversal to $\Sigma$. Set $D_{1}=\pi^{-1}_{1}(\Sigma)$.
Consider the statements:

$(1)_{\ell}$ $D_{\ell}\cup Y^{(\ell)}$ is normal crossings at each point of a
non terminal divisor $E$.

$(2)_{\ell}$ $\Sigma^{(\ell)}$ has rectified weight $-1$ and the remaining
irreducible components of $D_{\ell}$ have rectified weight $0$.

$(3)_{\ell}$ Each branch of $Y^{(\ell)}$ is non singular, intersects only one
divisor and is transversal to this divisor.

Statements $(i)_{\ell}$, $i=1,2,3$, $\ell\geq 1$, can be proved by induction
in $\ell$. Let $N$ be the maximal element of $\\{o(Y_{i},Y_{j}):i,j\in
I,i\not=j\\}$. The curve $Y$ is desingularized after $N$ blow ups. By
$(2)_{N}$, $\mathcal{R}^{\Sigma}_{Y}$ is a log elementary graph of smooth
type.

Let us prove the converse. We associate to each vertex $E$ of a rooted tree
with root $\Sigma$ its depth, the number of edges we have to cross in order to
go from $E$ to $\Sigma$. We call depth of the tree to the supreme of the
depths of its elements. Let $N$ be the depth of $\mathcal{R}^{\Sigma}_{Y}$.
Let $\pi:\widetilde{X}\to X$ be the minimal sequence of blow ups that
desingularizes $(Y,\Sigma)$. Set $D_{N}=\pi^{-1}(\Sigma)$. Let $Y^{(N)}$
[$\Sigma^{(N)}$] be the strict transform of $Y$ [$\Sigma$] by $\pi$. Hence
$Y^{(N)},\Sigma^{(N)},D_{N}$ verify $(i)_{N}$, $1\leq i\leq 3$. The divisors
of depth $N$ have valence $1$, hence weigth $-1$. Let us blow down these
divisors. Let $Y^{(N-1)},\Sigma^{(N-1)},D_{N-1}$ be the images of
$Y^{(N)},\Sigma^{(N)},D_{N}$ by the blow down. Statements
$(1)_{N-1},(2)_{N-1}$ are easily verified. If a branch of $Y^{(N-1)}$ was
singular, its strict transform would be singular or tangent to a divisor. If a
branch of $Y^{(N-1)}$ intersected two divisors, the blow up of its
intersection would create a divisor of positive rectified weigth. If a branch
of $Y^{(N-1)}$ was tangent to a divisor, its strict transform would intersect
two divisors. Hence we can iterate the procedure. By $(3)_{0}$ the branches of
$Y$ are smooth and transversal to $\Sigma$. ∎

######  Corollary 4.3.

Let $Y$ be a singular plane curve. Its resolution graph $\mathcal{R}_{Y}$ is
an elementary graph of smooth type if and only if all branches of $Y$ are non
singular.

###### Proof.

Let $Y$ be a singular plane curve. Let $\Sigma$ be a smooth curve transversal
to $Y$. The graph $\mathcal{R}_{Y}$ is an elementary graph of smooth type if
and only if $\mathcal{R}^{\Sigma}_{Y}$ is a log elementary graph of smooth
type. ∎

We shall depict the vertices of an elementary graph of smooth type using white
dots.

######  Example 4.4.

An elementary graph of smooth type. Let $Y_{i}=\\{y=\varphi_{i}(x)\\}$, $1\leq
i\leq 5$, where $\varphi_{1}(x)=x^{2}$, $\varphi_{2}(x)=x^{3}+x^{4}$,
$\varphi_{3}(x)=x^{3}-x^{4}$, $\varphi_{4}(x)=-x^{3}+x^{4}$ and
$\varphi_{5}(x)=-x^{3}-x^{4}$. Set $Y=\cup_{i=1}^{5}Y_{i}$.

Figure 1. The elementary graph $\mathcal{R}_{Y}$.

##  5\. Geometry of Continuous Fractions

Let $(X,o)$ be a germ of a smooth complex manifold. Let $\Sigma$ and $C$ be
two smooth transversal curves of $(X,o)$. Choose a system of local coordinates
$(x,y)$ of $(X,o)$ such that $\Sigma=\\{x=0\\}$ and $C=\\{y=0\\}$. Let $\Xi$
be the union of $\\{\Sigma,C\\}$ with the set of divisors that are created
when we desingularize the curves $\\{y^{k}-x^{n}=0\\}$, with $(k,n)=1$. We
will identify each element of $\Xi$ with each of its strict transforms.

Let $\mathbb{Q}_{+}$ be the set of positive integers. Set
$\overline{\mathbb{Q}}_{+}=\mathbb{Q}_{+}\cup\\{0,+\infty\\}$. The main
purpose of this section is to prove Lemma 5.1.

Let us give an invariant definition of $\Xi$. Set $X_{0}=X$, $D_{0}=\Sigma\cup
C$. Let $\pi_{n}:X_{n}\rightarrow X_{n-1}$ be the blow up of $X_{n-1}$ along
the singular locus of $D_{n-1}$. Set $D_{n}=\pi^{-1}_{n}(D_{n-1})$. Let
$\mathcal{R}_{n}$ be the dual graph of $D_{n}$. We will identify each
irreducible component of each $D_{n}$ with the corresponding strict transforms
by blow up. Let $\Xi_{n}$ be the set of equivalence classes of the vertices of
$\mathcal{R}_{n}$. Set $\Xi=\cup_{n}\Xi_{n}$. If $E,F\in\Xi$ have
representatives $E^{\prime},F^{\prime}$ such that $E^{\prime}\cap
F^{\prime}\not=\emptyset$, we will denote by $E\ast F$ the equivalence class
of the divisor created by the blow up of $E^{\prime}\cap F^{\prime}$. We will
identify an element of $\Xi$ with a convenient representative whenever this
identification does not create a serious ambiguity.

Let $E,F,G\in\Xi$. There is $m$ such that $E,F,G$ are vertices of
$\mathcal{R}_{m}$. We say that $F$ is between $E$ and $G$ , and write $E-F-G$,
if $F$ is a vertice of each path of $\mathcal{R}_{m}$ that connects $E$ and
$G$. Since the graphs $\mathcal{R}_{m}$ are linear, the relation $E<F$ if
$\Sigma-E-F$ is a total order of $\Xi$.

Given integers $a_{1},\ldots,a_{n}$, $a_{1}\geq 0$, $a_{2},\ldots,a_{n}\geq
1$, $n\geq 1$, we define recursively the rational number
$[a_{1},\ldots,a_{n}]$ by $[a_{1}]=a_{1}$ and
$[a_{1},\ldots,a_{n}]=a_{1}+1/[a_{2},\ldots,a_{n}]$, for $n\geq 2$. Each non
negative rational number admits an expansion in continuous fraction
$[a_{1},\ldots,a_{n}]$. Each positive rational number admits two expansions. A
positive integer $n$ admits the expansions $[n]$ and $[n-1,1]$. If
$\alpha\in\mathbb{Q}_{+}\setminus\mathbb{Z}$, ${\alpha}$ admits expansions
$[a_{1},\ldots,a_{n}]$ and $[a_{1},\ldots,a_{n}-1,1]$, with $a_{n}\geq 2$. We
set $[n,0]=+\infty$, for each non negative integer $n$.

If $a_{n}\geq 1$, we define the length of $[a_{1},\ldots,a_{n}]$ as
$|[a_{1},\ldots,a_{n}]|=a_{1}+\cdots+a_{n}$. We set $|0|=|+\infty|=0$. Given
$\alpha=[a_{1},\ldots,a_{k}]$,
$\beta=[b_{1},\ldots,b_{\ell}]\in\overline{\mathbb{Q}}_{+}$, we say that
$\alpha,\beta\in\overline{\mathbb{Q}}_{+}$ are related, and write
$\alpha\sim\beta$, if there is a non negative integer $n$ such that
$\alpha=[b_{1},\ldots,b_{\ell},n]$ or $\beta=[a_{1},\ldots,a_{k},n]$. If
$\beta=[a_{1},\ldots,a_{k},n]$, we define the convolution of $\alpha$ and
$\beta$ as the positive rational
$\alpha\ast\beta=\beta\ast\alpha=[a_{1},\ldots,a_{k},n+1]$.

(7) $\hbox{\rm If $\alpha\sim\beta$,}\qquad\alpha\sim\alpha\ast\beta\hbox{ \
\rm and }~{}|\alpha\ast\beta|=\max\\{|\alpha|,|\beta|\\}+1.$

Given $\alpha,\beta,\gamma\in\overline{\mathbb{Q}}_{+}$, we say that $\beta$
is between $\alpha$ and $\gamma$ , and write $\alpha-\beta-\gamma$, if
$\alpha<\beta<\gamma$ or $\gamma<\beta<\alpha$. Given
$R\subset\overline{\mathbb{Q}}_{+}$ and $\alpha,\beta\in R$, we say that
$\alpha$ and $\beta$ are contiguous in $R$, and write $\alpha R\beta$, if
there is no $\gamma\in R$ such that $\alpha~{}-~{}\gamma~{}-~{}\beta$.

Notice that $0\ast 1/n=[0]\ast[0,n]=[0,n+1]=1/(n+1)$,
$n\ast+\infty=[n]\ast[n,0]=[n,1]=n+1$ and

$0~{}-~{}\frac{1}{n+1}~{}-~{}\frac{1}{n},\;\;n~{}-~{}n+1~{}-~{}+\infty$

Let $\alpha,\beta,\gamma,\delta\in{\mathbb{Q}}_{+}$,
$\delta=[a_{1},\ldots,a_{k}]$, with $k\geq 1$. If
$\alpha~{}-~{}\beta~{}-~{}\gamma$,

$\frac{1}{\alpha}~{}-~{}\frac{1}{\beta}~{}-~{}\frac{1}{\gamma}\;\;\textrm{and}\;\;\left(a_{k}+\frac{1}{\alpha}\right)~{}-~{}\left(a_{k}+\frac{1}{\beta}\right)~{}-~{}\left(a_{k}+\frac{1}{\gamma}\right).$

Hence $[\delta,\alpha]-[\delta,\beta]-[\delta,\gamma]$. Therefore
$\alpha-\alpha\ast\beta-\beta$ whenever $\alpha\sim\beta$. Moreover,
$\alpha\sim\beta$ and $\alpha<\beta$ implies that

(8) $\alpha<\alpha\ast\beta<\beta.$

Relation (8) also holds if $\alpha=0$ or $\beta=+\infty$.

Let $\alpha,\beta\in{\mathbb{Q}}_{+}$. We say that $\alpha$ precedes $\beta$,
and we write $\alpha\prec\beta$, if $\beta=[a_{1},\ldots,a_{k}]$ and there are
integers $\ell,b$ such that $\alpha=[a_{1},\ldots,a_{\ell},b]$ and
$0\leq\ell\leq k-1,1\leq b\leq a_{\ell+1}$. The partial order $\prec$ induces
in $\mathbb{Q}_{+}$ a structure of tree with root $1$. We say that ${\alpha}$
immediately precedes $\beta$ if ${\alpha}\prec{\beta}$ and
$|{\beta}|=|{\alpha}|+1$.

Given two divisors $E,F$ of a resolution graph $\mathcal{R}$, we will say that
$E$ precedes $F$ and write $E\prec F$ if there is a sequence of divisors
$E=E_{0},E_{1},\ldots,E_{n}=F$ such that $E_{i}$ is created blowing up a point
of $E_{i-1}$, $1\leq i\leq n$.

######  Lemma 5.1.

There is a bijection $\alpha\mapsto E_{\alpha}$ from
$\overline{\mathbb{Q}}_{+}$ onto $\Xi$ such that

  * $(a)$

$\alpha<\beta$ if and only if $E_{\alpha}<E_{\beta}$.

  * $(b)$

If $\alpha\sim\beta$, $E_{\alpha}\ast E_{\beta}=E_{\alpha\ast\beta}$.

  * $(c)$

${\alpha}\prec{\beta}$ if and only if $E_{\alpha}\prec E_{\beta}$.

We will denote the inverse of $\alpha\mapsto E_{\alpha}$ by
$E\mapsto\alpha^{E}$.

###### Proof.

If $\mathcal{R}_{m}$ has vertices $E^{0}<E^{1}<\cdots<E^{r_{m}}$,
$\mathcal{R}_{m+1}$ has vertices $E^{0}<E^{0}\ast E^{1}<E^{1}<E^{1}\ast
E^{2}<E^{2}\cdots<E^{r_{m}}$. Since $\mathcal{R}_{0}$ has two vertices,
$\mathcal{R}_{m}$ has $2^{m}+1$ vertices, for each $m\geq 1$.

Set
$\overline{\mathbb{Q}}_{m}=\\{\alpha\in\overline{\mathbb{Q}}_{+}:|\alpha|\leq
m\\}$ for each $m\geq 0$. Each ${\alpha}\in\mathbb{Q}_{+}$ immediately
precedes two elements of $\mathbb{Q}_{+}$: $1$ immediately precedes $2$ and
$1/2$; if ${\alpha}=[a_{0},\ldots,a_{k}]$ and $a_{k}\geq 2$, ${\alpha}$
immediately precedes $[a_{0},\ldots,a_{k}+1]$, $[a_{0},\ldots,a_{k}-1,2]$ and
no other rational numbers. Hence $\\#\\{{\alpha}:|{\alpha}|=m\\}=2^{m-1}$ and
$\\#\overline{\mathbb{Q}}_{m}=2^{m}+1$, for each $m\geq 1$.

Let us show that, for each $m\geq 1$:

$(1)_{m}$ $\overline{\mathbb{Q}}_{m}=\\{\alpha_{0},\ldots,\alpha_{2^{m}}\\}$,
$\alpha_{0}<\cdots<\alpha_{2^{m}}$, $|\alpha_{i}|=m$ if $i$ odd,
$|\alpha_{i}|<m$ if $i$ even.

$(2)_{m}$ If $\alpha,\beta\in\overline{\mathbb{Q}}_{m}$ and
$\alpha\overline{\mathbb{Q}}_{m}\beta$, $\alpha\sim\beta$ .

$(3)_{m}$ There is a bijection ${\alpha}\mapsto E_{\alpha}$ from
$\overline{\mathbb{Q}}_{m}$ onto $\mathcal{R}_{m}$ such that $(a),(c)$ hold
for ${\alpha},{\beta}\in\overline{\mathbb{Q}}_{m}$ and $(b)$ holds for
${\alpha},{\beta}\in\overline{\mathbb{Q}}_{m-1}$.

Notice that $\overline{\mathbb{Q}}_{0}=\\{0,+\infty\\}$. If $|\alpha|=1$,
$\alpha=1$. Hence $\overline{\mathbb{Q}}_{1}=\\{0,1,+\infty\\}$. Since
$+\infty=[0,0]$, $0\sim+\infty$ and $0\ast+\infty=[0,1]=1$. By (7), $0\sim 1$
and $1\sim+\infty$. If $|\alpha|=2$, $\alpha=[0,2]=1/2$ or $\alpha=[1,1]=2$.
Since $0\ast 1=[0]\ast[0,1]=[0,2]$ and $1\ast+\infty=[1]\ast[1,0]=[1,1]=2$,
$0\sim 1/2$, $1/2\sim 1$, $1\sim 2$ and $2\sim+\infty$.

Assume that $(1)_{m}$ holds for $i=1,2$. By (7) and (8),

$|\alpha_{2i-2}\ast\alpha_{2i-1}|=|\alpha_{2i-1}\ast\alpha_{2i}|=m+1$

and

(9)
$\alpha_{2i-2}<\alpha_{2i-2}\ast\alpha_{2i-1}<\alpha_{2i-1}<\alpha_{2i-1}\ast\alpha_{2i}<\alpha_{2i},$

for $1\leq i\leq 2^{m-1}$. Hence $(1)_{m+1}$ holds and $(2)_{m+1}$ follows
from (7).

Set $E_{0}=\Sigma$ and $E_{+\infty}=C$. Assume that $(3)_{m}$ holds. By $(b)$,
$E$ has a unique extension to $\overline{\mathbb{Q}}_{m+1}$. By (9), $(a)$
holds for $\alpha,\beta\in\overline{\mathbb{Q}}_{m+1}$. By the definitions of
$"\alpha\ast\beta"$ and $"E\ast F"$, $(b)$ holds for
$\alpha,\beta\in\overline{\mathbb{Q}}_{m}$.

Assume $\alpha\in\mathbb{Q}_{m}$, $|\beta|=m+1$.

Assume $\alpha\prec\beta$. If $|\alpha|=m$, there is $\gamma\in\mathbb{Q}_{m}$
such that $\beta=\alpha\ast\gamma$. Hence $E_{\alpha}\prec E_{\beta}$.
Otherwise, there is $\delta$ such that $|\delta|=m$ and
$\alpha\prec\delta\prec\beta$. Hence $E_{\alpha}\prec E_{\delta}\prec
E_{\beta}$.

Assume $E_{\alpha}\prec E_{\beta}$. If $|\beta|=|\alpha|+1$, there is
$\gamma\in\mathbb{Q}_{m}$ such that $E_{\beta}=E_{\alpha}\ast E_{\gamma}$.
Hence $\beta=\alpha\ast\gamma$ and $\alpha\prec\beta$. Otherwise, there are
$\gamma,\delta\in\mathbb{Q}_{m}$ such that $E_{\beta}=E_{\gamma}\ast
E_{\delta}$ and $E_{\alpha}\prec E_{\gamma}$. Hence
$\alpha\prec\gamma\prec\gamma\ast\delta=\beta$. ∎

##  6\. Resolution Graphs of Cusp Type

We keep the notations of Section 4. Let $Y$ be a germ of an irreducible plane
curve of $(X,o)$, $Y\neq\Sigma$. Assume that if $Y$ is singular, $Y$ has
maximal contact with $C$ or $Y$ has maximal contact with $\Sigma$. Assume $Y$
has at most one Puiseux exponent. Given $\alpha>1$ [$\alpha<1,{\alpha}=1$] we
say that $Y$ is a cusp of exponent $\alpha$ relative to $(\Sigma,C)$ if
$o(Y,C)=\alpha$ [$o(Y,\Sigma)=1/\alpha,Y$ is transversal to $C$ and $\Sigma$].
We assume $\alpha=p/q$, where $p,q$ are positive integers and $(p,q)=1$.

Let $(x,y)$ be a system of local coordinates of $(X,o)$ such that
$\Sigma=\\{x=0\\}$ and $C=\\{y=0\\}$. If $Y$ is a cusp of exponent $\alpha$
and $\alpha\geq 1$ [$\alpha<1$], there is
$\varepsilon\in\mathbb{C}\\{x^{1/q}\\}$
[$\varepsilon\in\mathbb{C}\\{y^{1/p}\\}$] such that
$Y=\\{y=x^{\alpha}\varepsilon\\}$ [ $Y=\\{x=y^{\alpha}\varepsilon\\}$] and
$\varepsilon(0)\neq 0$. We say that $\varepsilon(0)$ is the coefficient of $Y$
relative to the system of local coordinates $(x,y)$. We say that $C$ has
exponent $+\infty$ and coefficient $0$. Notice that the coefficient of a cusp
depends on the system of local coordinates but the fact that two cusps have
the same exponent but different coefficients does not.

######  Lemma 6.1.

Let $Y$ $[Y_{i}]$ be a cusp of exponent $\alpha$ relatively to $(\Sigma,C)$,
$i=1,2$. Let $\widetilde{Y}$ $[\widetilde{Y}_{i}]$ be the strict transform of
$Y$ $[Y_{i}]$ by the minimal resolution of $Y\cup\Sigma\cup C$
$[Y_{i}\cup\Sigma\cup C]$. Then

  * $(a)$

$\widetilde{Y}$ is transversal to $E_{\alpha}$.

  * $(b)$

$\widetilde{Y}_{1}\cap\widetilde{Y}_{2}=\emptyset$ if and only if
$\widetilde{Y}_{1}$ and $\widetilde{Y}_{2}$ have different coefficients.

###### Proof.

By induction in $|\alpha|$. ∎

######  Lemma 6.2.

Let $(Y,\Sigma)$ be a logarithmic plane curve. Assume
$\mathcal{R}^{\Sigma}_{Y}$ linear. Assume that $\widetilde{\Sigma}$ has
valence $1$ and the terminal vertex of $\mathcal{R}^{\Sigma}_{Y}$ does not
have weight $-1$. Then

  * $(a)$

Each branch of $Y$ has at most one Puiseux exponent.

  * $(b)$

There is a smooth curve $C$ transversal to $\Sigma$ such that all singular
branches of $Y$ have maximal contact with $C$ or $\Sigma$.

  * $(c)$

If $Y_{i},Y_{j}$ are smooth branches of $Y$ transversal to $\Sigma$,
$o(Y_{i},Y_{j})\leq\tau_{C}(Y)$.

  * $(d)$

All branches of $Y$ are cusps relatively to $(\Sigma,C)$. If two branches of
$Y$ are cusps with the same exponent, they have different coefficients.

###### Proof.

Statement $(a)$ is well known. See for instance [1]. Let us prove $(b)$. Let
$\lambda_{i}$ be the exponent of each singular branch $Y_{i}$ of $Y$
transversal to $\Sigma$. If the curve $C$ does not exist, there are $i,j$ such
that $o(Y_{i},Y_{j})<\min\\{\lambda_{i},\lambda_{j}\\}$. After blowing up
$Y_{i}\cap Y_{j}$ $o(Y_{i},Y_{j})$-times, the curves $Y_{i}$ and $Y_{j}$ are
still singular and intersect different smooth points of $D_{o(Y_{i},Y_{j})}$.
Hence $\mathcal{R}^{\Sigma}_{Y}$ is not linear.

Assume that $(c)$ does not hold. Set $\tau=\tau_{C}(Y)$. Let $Y^{\prime}$
[$\Sigma^{\prime}$] be the germ of $Y^{(\tau)}$ [$E_{\tau}$] at
$Y^{(\tau)}_{i}\cap Y^{(\tau)}_{j}$. By Lemma 6.1, the strict transforms of
the singular branches of $Y$ only intersect divisors $E$ such that
$\alpha^{E}<\tau$. Hence, the curve $(Y^{\prime},\Sigma^{\prime})$ verifies
the conditions of Theorem 4.2. Therefore
$\mathcal{R}^{\Sigma^{\prime}}_{Y^{\prime}}$ and $\mathcal{R}^{\Sigma}_{Y}$
have a terminal vertex of weight $-1$.

Statement $(d)$ follows from Lemma 6.1 and the linearity of
$\mathcal{R}^{\Sigma}_{Y}$. ∎

Let $(Y,\Sigma)$ be a logarithmic plane curve such that
$\mathcal{R}^{\Sigma}_{Y}$ verifies the conditions of Lemma 6.2. Assume
$Y_{i}$ is a branch of $Y$ such that $o(Y_{i},C)>\tau_{C}(Y)$. By statement
$(c)$, $Y_{i}$ is smooth and $o(Y_{j},C)\leq\tau_{C}(Y)$, for all $j\neq i$.
Hence we can take $C=Y_{i}$. Under these conditions, the set of divisors of
$\mathcal{R}^{\Sigma}_{Y}$ is naturally identified with a subset of $\Xi$ (and
a subset of $\overline{\mathbb{Q}}_{+}$).

######  Lemma 6.3.

Assume $(Y,\Sigma)$ verifies the conditions of Lemma 6.2. Then:

  * $(a)$

$E_{\alpha}\in\mathcal{R}^{\Sigma}_{Y}$ if and only if $\alpha$ belongs to the
smallest subtree $R$ of $(\mathbb{Q}_{+},\prec)$ that contains $1$ and the
exponents of the branches of $Y$.

  * $(b)$

If $\alpha=[a_{1},\ldots,a_{k}]\in R$,

$\omega_{E_{\alpha}}=-1-\max\\{c,0:[a_{1},\ldots,a_{k},c]\in
R\\}-\max\\{c,0:[a_{1},\ldots,a_{k}-1,1,c]\in R\\}.$

###### Proof.

Let $Y$ [$Z$] be a cusp of type $[a_{1},\ldots,a_{k},c]$
[$[a_{1},\ldots,a_{k}-1,1,d]$]. After blowing up $D_{\ell}\cap Y^{(\ell)}$,
$\ell=0,1,\ldots,|{\alpha}|-1$, one creates the divisor $E_{\alpha}$.
Moreover, the cusps $Y^{(|{\alpha}|)},Z^{(|{\alpha}|)}$ intersect $E_{\alpha}$
at different points, $o(Y^{(|{\alpha}|)},E_{\alpha})=c$ and
$o(Z^{(|{\alpha}|)},E_{\alpha})=d$.

After blowing up $D_{\ell}\cap Y^{(\ell)}$,
$\ell=|{\alpha}|,\ldots,|{\alpha}|+c-1$ and $D_{\ell}\cap Z^{(\ell)}$,
$\ell=|{\alpha}|,\ldots,|{\alpha}|+d-1$, $Y^{(|{\alpha}|+c)}$ and
$Z^{(|{\alpha}|+d)}$ no longer intersect $E_{\alpha}$. Moreover, $E_{\alpha}$
has weight $-1-c-d$. ∎

Let $\mathcal{R}^{\Sigma}$ be a logarithmic resolution graph verifying the
conditions of Lemma 6.2. We can give an explicit embedding of the set of
vertices of $\mathcal{R}^{\Sigma}$ into $\mathbb{Q}_{+}\cup\\{0\\}$ by setting
$\alpha^{\Sigma}=0$, $\alpha^{C}=+\infty$ and
$\alpha^{E}=\alpha^{E^{\prime}}\ast\alpha^{E^{\prime\prime}}$ whenever we are
at a resolution step such that $\omega_{E}=-1$, $E^{\prime}\cap
E,E^{\prime\prime}\cap E\neq\emptyset$. Let $\tau(\mathcal{R}^{\Sigma})$ be
the smallest positive integer $\tau$ such that for each non divisor $E$ of
$\mathcal{R}^{\Sigma}$ with $\alpha^{E}\not\in\mathbb{Z}$, $\alpha^{E}<\tau$.

Assume that the resolution graph $\mathcal{R}_{Y}$ of $Y$ is linear, its root
has valence $1$ and its vertex does not have weight $-1$. Let $\Sigma$ be a
smooth curve transversal to $Y$. We obtain $\mathcal{R}^{\Sigma}_{Y}$ from
$\mathcal{R}_{Y}$ connecting $\Sigma$ to the root of $\mathcal{R}_{Y}$ and
setting $\omega_{\Sigma}=-2$. By Lemma 6.2, the set of divisors of $R_{Y}$ is
naturally identified with a subset of $\mathbb{Q}_{+}$. Set
$\tau(\mathcal{R}_{Y})=\tau(\mathcal{R}^{\Sigma}_{Y})$.

######  Definition 6.4.

Let $\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] be a [logarithmic] resolution
graph. We say that $\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] is a [log]elementary
graph of cusp type if:

  * $(a)$

$\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] is a linear graph and $\Sigma$ has
valence $1$.

  * $(b)$

$\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] is not a [log] elementary graph of
smooth type.

  * $(c)$

If an integer divisor $E$ of $\mathcal{R}$ [$\mathcal{R}^{\Sigma}$] intersects
a strict transform of a branch of the logarithmic curve,
$\alpha^{E}=\tau(\mathcal{R})$ [$\alpha^{E}=\tau(\mathcal{R}^{\Sigma})$].

Condition $(c)$ guarantees the uniqueness of the decomposition (see Definition
7.1 and Examples 7.6, 7.7).

######  Theorem 6.5.

Let $(Y,\Sigma)$ be a logarithmic plane curve. Its resolution graph is a log
elementary graph of cusp type if and only if:

  * $(a)$

there is a smooth curve $C$ transversal to $\Sigma$ and a system of local
coordinates $(x,y)$ such that $\Sigma=\\{x=0\\}$, $C=\\{y=0\\}$ and all
branches of $Y$ are cusps relatively to $(\Sigma,C)$.

  * $(b)$

If two branches of $Y$ have the same exponent, they have different
coefficients.

  * $(c)$

If $Y_{i}$ is a branch with integer exponent $\alpha$, $\alpha=\tau_{C}(Y)$.

  * $(d)$

$Y$ has a singular branch or a branch tangent to $\Sigma$.

###### Proof.

Assume $\mathcal{R}^{\Sigma}_{Y}$ is a log elementary graph of cusp type.
Statement $(d)$ follows from condition $(b)$ (of Definition 6.4) and Theorem
4.2. Statements $(a),(b)$ follow from condition $(a)$ and Lemma 6.2. Assume
that $Y$ has several smooth branches transversal to $\Sigma$. By Lemma 6.2 its
exponents are smaller than or equal to $\tau_{C}(Y)$. By condition $(c)$,
these exponents are equal to $\tau_{C}(Y)$.

Let us prove the converse. Condition $(a)$ follows from statements $(a),(b)$
and Lemmas 5.1,6.1. Condition $(b)$ follows from statement $(d)$ and Lemma
6.3. Condition $(c)$ follows from statement $(c)$. ∎

We will depict the vertices of an elementary graph of cusp type using black
dots.

######  Example 6.6.

An elementary graph of cusp type. Let $Y_{i}=\\{y=\varphi_{i}(x)\\}$, $1\leq
i\leq 3$, where $\varphi_{1}(x)=x^{3/2}$, $\varphi_{2}(x)=x^{5/3}$ and
$\varphi_{3}(x)=x^{5/2}$. Set $Y=\cup_{i=1}^{3}Y_{i}$.

Figure 2. The elementary graph $\mathcal{R}_{Y}$.

######  Corollary 6.7.

Let $Y$ be the germ of a plane curve. Its resolution graph is a elementary
graph of cusp type if and only if:

  * $(a)$

There is a smooth curve $\Sigma$ transversal to $Y$ and there is a smooth
curve $C$ transversal to $\Sigma$ such that all branches of $Y$ are cusps
relatively to $(\Sigma,C)$.

  * $(b)$

If two branches of $Y$ have the same exponent, they have different
coefficients.

  * $(c)$

If $Y_{i}$ is a branch with integer exponent $\alpha$, $\alpha=\tau_{C}(Y)$.

  * $(d)$

$Y$ has a singular branch.

##  7\. Decomposition of a Resolution Graph

######  Definition 7.1.

Let $\mathcal{G}$ be an elementary graph, $E$ a vertex of $\mathcal{G}$ and
$Z$ a branch that intersects $E$. Let $\mathcal{H}$ be a log elementary graph
with root $\Sigma$. We glue $\mathcal{G}$ with $\mathcal{H}$ at $Z$
withdrawing $Z$, identifying $\Sigma$ with $E$ and replacing $\omega_{E}$ by
$\omega_{E}+\omega_{\Sigma}+1$. The gluing of $\mathcal{G}$ with $\mathcal{H}$
at $Z$ is admissible if $E$ is not a black root and one of the following
conditions is verified:

  1. (a)

$E$ and $\Sigma$ have different colors.

  2. (b)

$E$ is black and the number of divisors connected to $E$ plus the number of
branches connected to $E$ is greater than $2$.

Let $\Lambda$ be a rooted tree with root $\phi$. Let
$(\mathcal{D}_{u})_{u\in\Lambda}$ be a family of elementary graphs. Assume all
log elementary with the possible exception of $\mathcal{D}_{\phi}$. Associate
to each vertex $u$ of $\Lambda$, $u\not=\phi$, its father $f(u)$, a divisor
$E_{u}$ of $\mathcal{D}_{f(u)}$ and a branch $Y_{u}$ that intersects $E_{u}$.
Assume that the map $u\mapsto Y_{u}$ is injective. We call gluing of
$(\mathcal{D}_{u})_{u\in\Lambda}$ to the graph we obtain gluing
$\mathcal{D}_{f(u)}$ with $\mathcal{D}_{u}$ at $Y_{u}$ for each
$u\in\Lambda\setminus\\{\phi\\}$.

We say that a resolution graph admits a decomposition into elementary graphs
if it is a gluing of elementary graphs. A decomposition into elementary graphs
of a resolution graph is called admissible if, the gluing of
$\mathcal{D}_{f(u)}$ with $\mathcal{D}_{u}$ at $Y_{u}$ is admissible for each
$u\in\Lambda\setminus\\{\phi\\}$.

######  Theorem 7.2.

A gluing of elementary graphs is a resolution graph.

###### Proof.

Let us prove the theorem by induction on the number of vertices of $\Lambda$.
The theorem holds if $\Lambda=\\{\phi\\}$. Let $u$ be a terminal vertex of
$\Lambda$, $u\not=\phi$. Set $\Lambda_{0}=\Lambda\setminus\\{u\\}$ and let $o$
be the intersection point of $Y_{u}$ and $E_{u}$. By the induction hypothesis,
the gluing of $(\mathcal{D}_{v})_{v\in\Lambda_{0}}$ is the resolution graph of
a plane curve $H$. Let $\pi:\widetilde{X}\rightarrow(X,\sigma)$ be the
resolution of $H$. Set ${D}=\pi^{-1}(\sigma)$. Let $\widetilde{H}$ be the
strict transform of $H$ by $\pi$. Let $\widetilde{H}_{u}$ be the germ
$\widetilde{H}$ at $o$. By Theorems 6.5, 4.2 there is a germ of a logarithmic
curve $(W,\Sigma)$ with resolution graph $\mathcal{D}_{u}$. We can assume that
$\Sigma$ is the germ of $E_{u}$ at $o$. Set $Y=\pi((\widetilde{H}\setminus
H_{u})\cup W)$. The resolution graph of $Y$ is obtained gluing the resolution
graph of $H$ and the resolution graph of $(W,\Sigma)$ through the
identification of $\Sigma$ with a germ of $E_{u}$. Let $\widetilde{E}$ be the
strict transform of $E_{u}$ by the sequence of blow ups that desingularizes
$(\widetilde{H}\setminus H_{u})\cup W$. If $\kappa$ denotes the number of blow
ups necessary to separate $W$ and $\Sigma$, ${\omega}_{\widetilde{E}}$ equals
$\omega_{E_{u}}-\kappa$. ∎

We call leading term of a Puiseux series $\sum_{\alpha}a_{\alpha}x^{\alpha}$
to $a_{\beta}x^{\beta}$, where $\beta$ is the smallest positive rational
$\gamma\in\mathbb{Q}$ such that $a_{\gamma}\neq 0$. If
$Z=\\{y=\sum_{\alpha}a_{\alpha}x^{\alpha}\\}$ is a germ of plane curve, set
$Z^{[\ell]}=\\{y=\sum_{{\alpha}\leq\ell}a_{\alpha}x^{j}\\}$.

######  Lemma 7.3.

Let $(Y,\Sigma)$ be a logarithmic plane curve, with $\Sigma$ smooth. There is
a curve $Y^{\phi}$ such that:

  1. (a)

the Puiseux expansion of each irreducible component $Y^{\phi}_{i}$ of
$Y^{\phi}$ is a truncation of the Puiseux expansion of some irreducible
component $Y_{i}$ of $Y$;

  2. (b)

the resolution graph of $(Y^{\phi},\Sigma)$ is a log elementary graph;

  3. (c)

the sequence of blow ups that desingularize $(Y,\Sigma)$ and
$(Y^{\phi},\Sigma)$ coincide up to the step when the inverse image of
$Y^{\phi}\cup\Sigma$ is normal crossings;

  4. (d)

let $\pi_{\phi}:X^{\phi}\to X$ [$\pi:\widetilde{X}\to X$] be the sequence of
blow ups that desingularizes $(Y^{\phi},\Sigma)$ [$(Y,\Sigma)$]. Let
$\widetilde{Y}$ be the strict transform of $Y$ by $\pi_{\phi}$. Set
$D^{\phi}=\pi^{-1}_{\phi}(\Sigma\cap Y)$. Let $\sigma$ be a point of
$D^{\phi}$ such that $D^{\phi}\cup\widetilde{Y}$ is not normal crossings at
${\sigma}$. Let $D^{\phi}_{\sigma}$ [$\widetilde{Y}_{\sigma}$] be the germ of
$D^{\phi}$ [$\widetilde{Y}]$ at $\sigma$. Then there is a truncation
$\widetilde{Y}^{\phi}_{\sigma}$ of $\widetilde{Y}_{\sigma}$ such that the
resolution graph of $(\widetilde{Y}^{\phi}_{\sigma},D^{\phi}_{\sigma})$ is an
elementary graph and the gluing of $\mathcal{R}^{\Sigma}_{Y^{\phi}}$ with
$\mathcal{R}^{D^{\phi}_{\sigma}}_{\widetilde{Y}^{\phi}_{\sigma}}$ at
$\widetilde{Y}^{\phi}_{\sigma}$ is admissible.

###### Proof.

Set $J=\\{i\in I:Y_{i}$ is transversal to $\Sigma\\}$ and $K=I\setminus J$.
For $i\in J$, set $\lambda_{i}$ as the first Puiseux exponent of $Y_{i}$, if
$Y_{i}$ is singular, and $\lambda_{i}=+\infty$ otherwise. If $Y_{i}$ is smooth
for each $i\in J$, let $(x,y)$ be a system of local coordinates such that
$\Sigma=\\{x=0\\}$. Assume that there is $i\in J$ such that $Y_{i}$ is
singular. Let $\rho$ be the supremum of the set of integers $\ell$ such that
there is a smooth curve $C$ that verifies, for each $i\in J$,
$o(Y_{i},C)\geq\ell$ or $o(Y_{i},C)=\lambda_{i}$. Choose a smooth curve $C$
such that $C$ is transversal to $\Sigma$ and, for each $i\in J$,
$o(Y_{i},C)\geq\rho$ or $o(Y_{i},C)=\lambda_{i}$. Let $(x,y)$ be a system of
local coordinates such that $\Sigma=\\{x=0\\}$ and $C=\\{y=0\\}$. Let
$\tau_{0}$ be the supremum of $\lambda_{i}$ such that $i\in J$, $Y_{i}$ is
singular and $\lambda_{i}\leq\rho$. Let $\tau$ be the smallest positive
integer bigger than $\tau_{0}$.

$(A)$ Assume $K=\emptyset$ and $\tau=1$. Let $q_{i}$ be the supremum of the
set of integers $\ell$ such that $Y^{[\ell]}_{i}$ is smooth and
$o(Y^{[\ell]}_{i},Y^{[\ell]}_{j})\in\mathbb{Z}\cup\\{+\infty\\}$ for each
$j\in I$. Set $Y^{\phi}_{i}=Y^{[q_{i}]}_{i}$ for each $i\in I$. Set
$Y^{\phi}=\cup_{i\in I}Y^{\phi}_{i}$.

$(B)$ Assume $K\neq\emptyset$ or $\tau\geq 2$. If $i\in K$ and
$Y_{i}=\\{x=\psi_{i}(y)\\}$, set
$Y^{\phi}_{i}=\\{x=\widetilde{\psi}_{i}(y)\\}$, where $\widetilde{\psi}_{i}$
is the leading term of $\psi_{i}$. Assume $i\in J$ and
$Y_{i}=\\{y=\varphi_{i}(x)\\}$. If $o(Y_{i},C)>\tau$, set $Y^{\phi}_{i}=C$.
Otherwise, set $Y^{\phi}_{i}=\\{y=\widetilde{\varphi}_{i}(x)\\}$, where
$\widetilde{\varphi}_{i}$ is the leading term of $\varphi_{i}$. Set
$Y^{\phi}=\cup_{i\in I}Y^{\phi}_{i}$.

By construction, $(Y^{\phi},\Sigma)$ verifies condition $(a)$. In case $(A)$
[$(B)$], $(Y^{\phi},\Sigma)$ verifies the conditions of Theorem 4.2 [Theorem
6.5 ], hence statement $(b)$ holds.

Statement $(c)$ follows from Theorems 2.1 and 2.3.

Let us show that statement $(d)$ holds.

Assume $(A)$. Let $\widetilde{Y_{i}}$ $[\widetilde{Y^{\phi}_{i}}]$ be the
strict transform of $Y_{i}$ $[Y^{\phi}_{i}]$ by $\pi_{\phi}$. Choose $\sigma$
such that $D^{\phi}\cap\widetilde{Y}$ is not normal crossings at $\sigma$. By
the definition of the $q_{i}$’s, there is $i\in I$ such that
$\sigma\in\widetilde{Y_{i}}$, $Y^{[q_{i}+1]}_{i}$ is singular and
$\widetilde{Y_{i}}$ is either singular or tangent to $D^{\phi}$ at $\sigma$.
Hence $\widetilde{Y^{\phi}_{\sigma}}$ verifies condition $(B)$.

Assume $(B)$. Let $\widetilde{C}$ be the strict transform of $C$ by
$\pi_{\phi}$. By the definition of admissibility, we only need to check what
happens at the point $\sigma$ such that $\widetilde{C}\cap
D^{\phi}=\\{\sigma\\}$. Set $J_{\phi}=\\{i\in J:Y^{\phi}_{i}=C\\}$,
$Y_{\phi}=\cup_{i\in J_{\phi}}Y_{i}$. Notice that the germ at $\sigma$ of the
strict transform of $Y$ by $\pi_{\phi}$ equals the strict transform of
$Y_{\phi}$ by $\pi_{\phi}$. If $J_{\phi}=\emptyset$ or $Y_{\phi}$ is smooth,
nothing is glued to $\mathcal{R}^{\Sigma}_{Y^{\phi}}$ at $\sigma$ and there is
nothing to prove. Assume $Y_{\phi}$ is singular. By the construction of
$Y^{\phi}$, in particular the definitions of $\rho$ and $\tau$, the minimum of
the set $\\{o(Y_{i},Y_{j}):~{}i,j\in J_{\phi}\\}$ is an integer. Hence
$\widetilde{Y}^{\phi}_{\sigma}$ verifies condition $(A)$. ∎

######  Theorem 7.4.

The resolution graph of a logarithmic plane curve $(Y,\Sigma)$, with $\Sigma$
smooth, admits one and only one admissible decomposition into elementary
graphs.

###### Proof.

Let $g_{i}$ be the number of Puiseux pairs of the branch $Y_{i}$ of $Y$, $i\in
I$. Set $h_{i}=g_{i}$ if $Y_{i}$ is transversal to $\Sigma$. Otherwise, set
$h_{i}=g_{i}+1/2$. Set $h_{Y}=\max\\{h_{i}:i\in I\\}$. Let $\\#_{Y}$ be the
cardinality of $I$. Let us construct an admissible decomposition into
elementary graphs $(\mathcal{D}_{v})_{v\in\Lambda}$ of
$\mathcal{R}^{\Sigma}_{Y}$ by induction on $h_{Y}$ and $\\#_{Y}$.

Let $Y^{\phi}$ be the curve constructed in Lemma 7.3. We will follow the
notations of this Lemma.

If ${\sigma}\in D^{\phi}$ and $D^{\phi}\cup\widetilde{Y}$ is not normal
crossings at ${\sigma}$, $h_{\widetilde{Y}_{{\sigma}}}<h_{Y}$ or
$\\#_{\widetilde{Y}_{{\sigma}}}<\\#_{Y}$. Hence
$(\widetilde{Y}_{{\sigma}},D^{\phi}_{{\sigma}})$ admits an admissible
decomposition into log elementary graphs. Since $\mathcal{R}^{\Sigma}_{Y}$ is
the gluing of $\mathcal{R}^{\Sigma}_{Y^{\phi}}$ and the graphs
$\mathcal{R}^{D^{\phi}_{\sigma}}_{\widetilde{Y}^{\phi}_{\sigma}}$, we obtain
an admissible decomposition into elementary graphs of
$\mathcal{R}^{\Sigma}_{Y}$.

Let $(\mathcal{E}_{v})_{v\in\Lambda^{\prime}}$ be another admissible
decomposition of $\mathcal{R}^{\Sigma}_{Y}$. Let us show by induction on
$h_{Y}$ and $\\#_{Y}$ that the two decompositions are equal. It is enough to
show that $\mathcal{D}_{\phi}=\mathcal{E}_{\phi}$.

Assume that $\mathcal{D}_{\phi}$ is of cusp type and $\mathcal{E}_{\phi}$ is
of smooth type. Let $(E,\Sigma)$ [$(D,\Sigma)$] be a logarithmic plane curve
such that $\mathcal{R}^{\Sigma}_{E}=\mathcal{E}_{\phi}$
[$\mathcal{R}^{\Sigma}_{D}=\mathcal{D}_{\phi}$]. let $\pi_{E}:X^{E}\to X$
[$\pi_{D}:X^{D}\to X$] be the sequence of blow ups that desingularizes
$(E,\Sigma)$ [$(D,\Sigma)$]. There is a subtree $R$ of
$(\mathbb{Q}_{+},\prec)$ such that we can identify the set of vertices of
$\mathcal{D}_{\phi}$ with $\\{E_{\alpha}:\alpha\in R\\}$. Let $m$ be the
biggest integer of $R$. By condition $(c)$ of Definition 6.4, the vertices
$E_{1},\ldots,E_{m}$ of $\mathcal{D}_{\phi}$ are also vertices of
$\mathcal{E}_{\phi}$ and the strict transform of $E$ by $\pi_{D}$ intersects
$E_{m}$ at a smooth point of $\pi^{-1}_{D}(\Sigma)$. Hence, The strict
transform of $D$ by $\pi_{E}$ intersects $E_{m}$ at a singular point of
$\pi^{-1}_{E}(\Sigma)$. Therefore the decomposition
$(\mathcal{E}_{v})_{v\in\Lambda^{\prime}}$ is not admissible and
$\mathcal{D}_{\phi}$ and $\mathcal{E}_{\phi}$ must be of the same type. Let
$G$ be a vertex of $\mathcal{E}_{\phi}$ and $F$ a vertex of
$\mathcal{R}^{\Sigma}_{Y}$. Since $\mathcal{E}_{\phi}$ is a resolution graph,

(10) $\textrm{if}\;F\prec G,\;F\;\textrm{is a vertex of}\;\mathcal{E}_{\phi}.$

Assume that there is a vertex of $\mathcal{D}_{\phi}$ that is not a vertex of
$\mathcal{E}_{\phi}$. Let $\gamma$ be the biggest element of $R$ such that
$E_{\gamma}$ is a vertex of $\mathcal{E}_{\phi}$. Let $n$ be the smallest
integer bigger than or equal to $\gamma$. Then $\gamma=n$. Otherwise
$E_{n}\prec E_{\gamma}$ and $\mathcal{E}_{\phi}$ would not be a resolution
graph by (10). Since $\mathcal{D}_{\phi}$ is linear and $\mathcal{D}_{\phi}$
properly contains $\mathcal{E}_{\phi}$, $E$ has a smooth branch $E^{\prime}$
such that the strict transform of $E^{\prime}$ by $\pi_{E}$ intersects
$E_{n}$. Therefore $(D,\Sigma)$ does not verify condition $(c)$ of Definition
6.4. Hence the decomposition $(\mathcal{E}_{v})_{v\in\Lambda^{\prime}}$ is not
admissible.

If there is a vertex $E$ of $\mathcal{E}_{\phi}$ that is not a vertex of
$\mathcal{D}_{\phi}$, the same argument shows that
$(\mathcal{D}_{v})_{v\in\Lambda}$ is not admissible.

Assume that $\mathcal{D}_{\phi}$ is of smooth type. Hence $\mathcal{E}_{\phi}$
is also of smooth type. Assume $\mathcal{D}_{\phi}\neq\mathcal{E}_{\phi}$.
Hence we can assume that there is a vertex $E$ of $\mathcal{D}_{\phi}$ and
$\mathcal{E}_{\phi}$ is connected to a vertex $F$ of $\mathcal{E}_{\phi}$ that
is not a vertex of $\mathcal{D}_{\phi}$. Let $\mathcal{F}$ be the subtree of
$\mathcal{E}_{\phi}$ with vertices the divisors $D$ of $\mathcal{E}_{\phi}$
such that $D=F$ or $F\prec D$. Let us reset $\omega_{E}$ in such a way that
$\omega_{E}$ plus the set of vertices of $\mathcal{F}$ connected to $E$ equal
$-1$. We obtain in this way a resolution graph $(\mathcal{F},E)$. Since
$\mathcal{E}_{\phi}$ is of smooth type, $\mathcal{F}$ is of smooth type. Hence
the gluing of $(\mathcal{D}_{v})_{v\in\Lambda}$ is not admissible. ∎

######  Corollary 7.5.

The resolution graph of a plane curve admits one and only one admissible
decomposition into elementary graphs.

######  Example 7.6.

Let $Y_{i}=\\{y=\varphi_{i}(x)\\}$, $i=1,2$, where
$\varphi_{1}(x)=x^{2}+x^{7/2}$ and $\varphi_{2}(x)=x^{7/2}$. We have
$Y^{\phi}_{i}=\\{y=\varphi^{\phi}_{i}(x)\\}$, $i=1,2$, with
$\varphi^{\phi}_{1}(x)=x^{2}$ and $\varphi^{\phi}_{2}(x)=0$. Set $Y=Y_{1}\cup
Y_{2}$. See Figure 3. The decomposition (C) is not admissible since the
resolution graph of $Y_{1}\cup Y^{\phi}_{2}$ does not verify condition (c) of
Definition 6.4.

(a) The graph $\mathcal{R}_{Y}$

(b) The canonical decomposition of $\mathcal{R}_{Y}$

(c) Another decomposition of $\mathcal{R}_{Y}$

Figure 3. Example 7.6.

######  Example 7.7.

Let $Y_{i}=\\{y=\varphi_{i}(x)\\}$, $1\leq i\leq 4$, where
$\varphi_{1}(x)=x^{3/2}$, $\varphi_{2}(x)=x^{5/2}$,
$\varphi_{3}(x)=x^{2}+x^{4}$ and $\varphi_{4}(x)=x^{2}-x^{4}$. We have
$Y^{\phi}_{i}=\\{y=\varphi^{\phi}_{i}(x)\\}$, $1\geq i\geq 4$, with
$\varphi^{\phi}_{1}(x)=\varphi_{1}(x)=x^{3/2}$, $\varphi^{\phi}_{2}(x)=0$ and
$\varphi^{\phi}_{3}(x)=\varphi^{\phi}_{4}(x)=x^{2}$. Set
$Y=\cup_{i=1}^{4}Y_{i}$. See Figure 4. The decomposition (C) is not admissible
since the resolution graph of $Y_{1}\cup Y^{\phi}_{3}\cup Y_{2}$ does not
verify condition (c) of Definition 6.4.

(a) $\mathcal{R}_{Y}$

(b) The canonical decomposition of $\mathcal{R}_{Y}$

(c) Another decomposition of $\mathcal{R}_{Y}$

Figure 4. Example 7.7.

## References

  * [1] E. Brieskorn and H. Knorrer - Plane Algebraic Curves, Birkhauser (1981).
  * [2] J. Cabral, O. Neto and P.C. Silva - On the local fundamental group of a plane curve, in preparation.
  * [3] M. Kashiwara, Quasi-unipotent constructible sheaves, J. Fac. Sci. Univ. Tokyo Sec. IA 28 (1982) (3) pp. 757-773.
  * [4] O. Neto and P.C. Silva The fundamental group of an algebraic link, C. R. Acad. Sci. Paris, Ser. I. 340 (2005), 141-146.
  * [5] C.T.C. Wall - Singular points of Plane Curves, London math Society (2004).
  * [6] Yang Jingen, Curve Singularities and graphs, Acta Mathematica Sinica 1990, Vol. 6, No 1, pp. 87-96.
  * [7] H. Laufer, Normal two-dimensional singularities, Ann. of Math. study 71, Princeton Univ. Press, 1971.

João Cabral CMAF, Universidade de Lisboa and Departamento de Matemática,
Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa

Orlando Neto CMAF and Faculdade de Ciências, Universidade de Lisboa

Pedro C. Silva Centro de Estudos Florestais and Departamento de Ciências e
Engenharia de Biossistemas, Inst. Superior de Agronomia, Universidade de
Lisboa

This research was partially supported by FEDER and FCT-Plurianual 2011.


# Uncertainty relations for any multi observables

Jinchuan Hou, Kan He  College of Mathematics, Institute of Mathematics,
Taiyuan University of Technology, Taiyuan 030024, P. R. China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Uncertainty relations describe the lower bound of product of standard
deviations of observables. By revealing a connection between standard
deviations of quantum observables and numerical radius of operators, we
establish a universal uncertainty relation for any $k$ observables, of which
the formulation depends on the even or odd quality of $k$. This universal
uncertainty relation is tight at least for the cases $k=2n$ and $k=3$. For two
observables, the uncertainty relation is exactly a simpler reformulation of
Schrödinger’s uncertainty principle.

PACS 2010. 03.65.Ta, 03.65.Db, 02.30.Tb

Key words and phrases. quantum states, uncertainty principle, observables,
deviations

##  1\. Introduction

In the past ninety years, the theory of quantum mechanics was applied in lots
of other sciences, including Information Science, Chemistry and Biology (Ref.
[2, 3, 9]). The uncertainty principle, discovered first by Heisenberg in 1927
(Ref. [7]), is often considered as one of the most important topics of quantum
theory (Ref. [8, 13]) and can be linked to quantum entanglement and other
important topics (Ref. [1, 10]). Heisenberg’s uncertainty principle says that

$None$ $\Delta_{q}\Delta_{p}\geq\frac{\hbar}{2},$

where $\Delta_{q}$ and $\Delta_{p}$ denote standard deviations of the position
$\hat{q}$ and momentum $\hat{p}$ respectively,
$\hbar=\frac{1}{2}|\langle\hat{q}\hat{p}-\hat{p}\hat{q}\rangle|$ is the
reduced Planck constant. Recalled that a quantum system can be simulated in a
complex Hilbert space $H$ with the inner product $\langle\cdot|\cdot\rangle$
and a pure state is described by a unit vector $|x\rangle$. Quantum
observables for a state $|x\rangle$ are self-adjoint operators on $H$ with
domain containing $|x\rangle$ (Ref. [13]). The value of observable $A$ for the
pure state $|x\rangle$ is $\langle A\rangle=\langle x|A|x\rangle$. In 1929,
Robertson generalized Heisenberg’s uncertainty principle, which says that, for
observables $A,B$ and pure state $|x\rangle$,

$None$ $\Delta_{A}\Delta_{B}\geq\frac{1}{2}|\langle[A,B]\rangle|.$

Where $[A,B]=AB-BA$ is the Lie product of $A$ and $B$,

$\Delta_{A}=\sqrt{\langle A^{2}\rangle-\langle A\rangle^{2}}\ {\rm and}\
\Delta_{B}=\sqrt{\langle B^{2}\rangle-\langle B\rangle^{2}}$

are the standard deviations of $A$ and $B$, respectively [11]. Schrödinger
gave a uncertainty principle, which is sharper than Robertson’s and asserts
that

$None$
$\Delta_{A}\Delta_{B}\geq\sqrt{\frac{1}{4}|\langle[A,B]\rangle|^{2}+|\frac{1}{2}\langle\\{A,B\\}\rangle-\langle
A\rangle\langle B\rangle|^{2}},$

where $\\{A,B\\}=AB+BA$ is the Jordan product of $A$ and $B$ [12].
Schrödinger’s uncertainty principle holds for mixed state, too. Recall that a
mixed state $\rho$ is a positive operator on $H$ with trace 1. Then the value
of observable $A$ for the state $\rho$ is $\langle A\rangle={\rm Tr}(A\rho)$
and the standard deviation of $A$ is $\Delta_{A}=\sqrt{\langle
A^{2}\rangle-\langle A\rangle^{2}}=\sqrt{{\rm Tr}(A^{2}\rho)-{\rm
Tr}(A\rho)^{2}}$. Here we assume that both ${\rm Tr}(A\rho)$ and ${\rm
Tr}(A^{2}\rho)$ are finite.

What happens for multi observables?

There is a simple way to get certain uncertainty relation from the uncertainty
principles in (1.2) or (1.3). For example, let $A,B,C$ be three observables,
then by applying (2) one gets

$None$
$\Delta_{A}^{2}\Delta_{B}^{2}\Delta_{C}^{2}\geq\frac{1}{8}|\langle[A,B]\rangle\langle[B,C]\rangle\langle[A,C]\rangle|.$

But (1.4) is not sharp enough.

Let $A=\hat{q},B=\hat{p}$ and $C=\hat{r}=-\hat{p}-\hat{q}$; then
$[p,q]=[q,r]=[r,p]=\frac{\hbar}{i}$, and thus (1.4), together with (1.1),
gives

$\Delta_{q}^{2}\Delta_{p}^{2}\Delta_{r}^{2}\geq(\frac{\hbar}{2})^{3}.$

However, in [15], a tight uncertainty relation is given that

$None$
$\Delta_{q}^{2}\Delta_{p}^{2}\Delta_{r}^{2}\geq(\tau\frac{\hbar}{2})^{3}$

with $\tau=\frac{2}{\sqrt{3}}>1$.

This also happens for Pauli matrices $X,Y,Z$. As $[X,Y]=2iZ$, one has
$\frac{1}{2}|\langle[X,Y]\rangle|=|\langle Z\rangle|$. Similarly
$\frac{1}{2}|\langle[X,Z]\rangle|=|\langle Y\rangle|$ and
$\frac{1}{2}|\langle[Y,Z]\rangle|=|\langle X\rangle|$. Thus by (1.4)

$\Delta_{X}^{2}\Delta_{Y}^{2}\Delta_{Z}^{2}\geq|\langle X\rangle\langle
Y\rangle\langle Z\rangle|.$

But it was announced by S.-M. Fei that

$None$
$\Delta_{X}^{2}\Delta_{Y}^{2}\Delta_{Z}^{2}\geq\frac{8}{3\sqrt{3}}{|\langle
X\rangle\langle Y\rangle\langle Z\rangle|}.$

This inequality is also tight and achieves “=” at
$\rho=\frac{1}{2}(I+\frac{1}{\sqrt{3}}X+\frac{1}{\sqrt{3}}Y+\frac{1}{\sqrt{3}}Z)$.

Therefore, to obtain uncertainty relations for multi observables that are
sharp enough, one needs new approaches. Let $A_{1},A_{2},\ldots,A_{k}$ be any
$k$ observerbles of a quantum system. The purpose of this paper is to
establish a lower bound of $\Delta_{A_{1}}\Delta_{A_{2}}\cdots\Delta_{A_{k}}$
in terms of $\langle A_{i}A_{j}\rangle,\langle A_{i}\rangle\langle
A_{j}\rangle$ and $\langle A_{j}^{2}\rangle$. For the case when $k=2$, the
uncertainty relation is equivalent to Schrödinger’s uncertainty principle that
is tight and has a simpler representation. For the case when $k=3$, we show
that the untainty relation is tight by taking Pauli matrices as observables.
All proofs of the main result and the lemmas will be presented in the appendix
section.

##  2\. uncertainty relations for multi observables

Our main idea is based on the following observation, which establishes a
formula to connect the standard deviation of a quantum observable $A$ of a
state $|x\rangle$ to the norm as well as the numerical radius of
$[A,|x\rangle\langle x|]$, the Lie product of $A$ and the rank one projection
$|x\rangle\langle x|$.

Let $T$ be a bounded linear operator acting on a complex Hilbert space $H$.
The numerical range of $T$ is the set $W(T)=\\{\langle
x|T|x\rangle\,:\,|x\rangle\in H,\||x\rangle\|=1\\},$ and the numerical radius
of $T$ is $w(T)=\sup\\{|\lambda|\,:\,\lambda\in W(T)\\}.$ The topic of
numerical range and numerical radius plays an important role in mathematics
and is applied into many areas (Ref. [4, 5, 6]). Denote by $\|T\|$ the
operator norm of $T$.

Lemma 2.1. Let $|x\rangle$ be a pure state and $A$ an observable for it. Then

$\Delta_{A}=\|[A,|x\rangle\langle x|]\|=w([A,|x\rangle\langle x|]).$

By Lemma 2.1, for any observables $A_{1},A_{2},\ldots,A_{k}$ for a pure state
$|x\rangle$,

$None$ $\Pi_{j=1}^{k}\Delta_{A_{j}}=\Pi_{j=1}^{k}\|[A_{j},|x\rangle\langle
x|]\|\geq\|\Pi_{j=1}^{k}[A_{j},|x\rangle\langle x|]\|\geq
w(\Pi_{j=1}^{k}[A_{j},|x\rangle\langle x|]).$

Note that, the value of $\Pi_{j=1}^{k}\Delta_{A_{j}}$ does not depend on the
order arrange of observables but $w(\Pi_{j=1}^{k}[A_{j},|x\rangle\langle x|])$
does. Therefore, the inequality (2.1) can be sharped to

$None$
$\Pi_{j=1}^{k}\Delta_{A_{j}}\geq\max_{\pi}w(\Pi_{j=1}^{k}[A_{\pi(j)},|x\rangle\langle
x|]),$

where the maximum is over all permutations $\pi$ of $(1,2,\ldots,k)$. Thus the
question of establishing an uncertainty relation for $k$ observales is reduced
to the question of calculating the numerical radius of the operator

$None$ $D_{k}^{(\pi)}=\Pi_{j=1}^{k}[A_{\pi(j)},|x\rangle\langle x|],$

which is an operator of rank $\leq 2$.

The exact value of $w(D_{k}^{(\pi)})$ is computable and we can establish an
uncertainty relation for any multi observables by (2.2). For simplicity, and
with no loss of generality, we state our results only for $\pi={\rm id}$.

One may ask why do not work on the stronger inequality

$\Pi_{j=1}^{k}\Delta_{A_{j}}\geq\|\Pi_{j=1}^{k}[A_{j},|x\rangle\langle x|]\|?$

In fact, as we show in the Appendix section, this stronger inequality leads to
weaker uncertainty relations. So the numerical radius is the better choice.

The following is our main result, here we agree on $\Pi_{j\in\Lambda}a_{j}=1$
if $\Lambda=\emptyset$. It is surprising that our uncertainty relation for any
$k$ observables has different formulation depending on the even or odd quality
of the integer $k$.

Theorem 2.2. Let $A_{1},A_{2},\ldots,A_{k}$ with $k\geq 2$ be observables.

(1) If $k=2n$, then

$None$ $\begin{array}[]{rl}&\Pi_{j=1}^{2n}\Delta_{A_{j}}\\\
\geq&\frac{1}{2}(\Pi_{j=1}^{n-1}|\langle A_{2j}A_{2j+1}\rangle-\langle
A_{2j}\rangle\langle A_{2j+1}\rangle|)(|\langle A_{1}A_{2n}\rangle-\langle
A_{1}\rangle\langle A_{2n}\rangle|+\Delta_{A_{1}}\Delta_{A_{2n}}).\end{array}$

(2) If $k=2n+1$, then, identifying $2n+2$ with $[(2n+2)$ mod $(2n+1)]=1$,

$None$ $\begin{array}[]{rl}&\Pi_{j=1}^{2n+1}\Delta_{A_{j}}\\\
\geq&\frac{1}{2}[2\Pi_{j=1}^{2n+1}|\langle A_{j}A_{j+1}\rangle-\langle
A_{j}\rangle\langle A_{j+1}\rangle|\\\
&+\Delta_{A_{1}}^{2}\Pi_{j=1}^{n}|\langle A_{2j}A_{2j+1}\rangle-\langle
A_{2j}\rangle\langle A_{2j+1}\rangle|^{2}\\\
&+\Delta_{A_{2n+1}}^{2}\Pi_{j=1}^{n}|\langle A_{2j-1}A_{2j}\rangle-\langle
A_{2j-1}\rangle\langle A_{2j}\rangle|^{2}]^{\frac{1}{2}}.\end{array}$

Obviously, “=” holds if and only if

$None$ $\Pi_{j=1}^{k}\|[A_{j},|x\rangle\langle
x|]\|=\|\Pi_{j=1}^{k}[A_{j},|x\rangle\langle
x|]\|=w(\Pi_{j=1}^{k}[A_{j},|x\rangle\langle x|]).$

Thus the uncertainty relation is tight if Eq.(2.6) holds for some observerbles
$A_{1},A_{2},\ldots,A_{k}$ and some state. This is the case as will be
illustrated in Section 4.

We remak that Theorem 2.2 holds for any state $\rho$ with $|{\rm
Tr}(A_{j}\rho)|<\infty$ and ${\rm Tr}(A_{j}^{2}\rho)<\infty$,
$j=1,2,\ldots,k$. To see this, denote by ${\mathcal{C}}_{2}(H)$ be the
Hilbert-Schimit class in $H$, which is a Hilbert space with inner product
$\langle T,S\rangle={\rm Tr}(T^{\dagger}S)$. Then, a positive operator $\rho$
is a state if and only if $\sqrt{\rho}$ is a unit vector in
${\mathcal{C}}_{2}(H)$. For a self-adjoint operator $A$ on $H$, define a
linear operator $L_{A}$ on ${\mathcal{C}}_{2}(H)$ by $L_{A}T=AT$ if ${\rm
Tr}(T^{\dagger}A^{2}T)<\infty$. It is clear that $L_{A}$ is self-adjoint as
$L_{A}^{\dagger}=L_{A^{\dagger}}=L_{A}$. Note that

$\langle A\rangle={\rm
Tr}(A\rho)=\langle\sqrt{\rho}|L_{A}|\sqrt{\rho}\rangle=\langle L_{A}\rangle$

and thus

$\Delta_{A}=\Delta_{L_{A}},\quad\langle L_{A}L_{B}\rangle=\langle
L_{AB}\rangle=\langle AB\rangle.$

Then, Theorem 2.2 is true by applying (2.1) to
$L_{A_{1}},L_{A_{2}},\ldots,L_{A_{k}}$ and the pure state
$|\sqrt{\rho}\rangle$.

Before to see the uncertainty relations presented by theorem 2.2 is sharper
than those obtained by Heisenberg’s uncertainty principle (1.2) and
Schrödinger’s uncertainty principle (1.3), we illustrate some application of
Theorem 2.2 for the cases $k=2$.

##  3\. The case of $k=2$: a reformulation of Schrödinger’s principle

Applying Theorem 2.2 (1) to the case when $k=2$, the following result is
immediate.

Theorem 3.1. Let $A$ and $B$ be observables for a state. Then

$None$ $\Delta_{A}\Delta_{B}\geq|\langle AB\rangle-\langle A\rangle\langle
B\rangle|,$

which is equivalent to Schrödinger’s uncertainty principle.

The expression of inequality is quite simpler than that of Schrödinger’s
uncertainty principle. We show that (3.1) is in fact equivalent to
Schrödinger’s uncertainty principle (1.3).

To check it, write $\langle A\rangle\langle B\rangle=r$ and $\langle
AB\rangle=s+it$, where $s,t,r\in\mathbb{R}$. Then $\langle BA\rangle=s-it$. A
simple computation gives

$|\langle AB\rangle-\langle A\rangle\langle B\rangle|=\sqrt{(s-r)^{2}+t^{2}},$
$\sqrt{\frac{1}{4}|\langle[A,B]\rangle|^{2}+|\frac{1}{2}\langle\\{A,B\\}\rangle-\langle
A\rangle\langle B\rangle|^{2}}=\sqrt{(s-r)^{2}+t^{2}}$

and

$\frac{1}{2}|\langle[A,B]\rangle|=|t|.$

So, we get

$None$ $\begin{array}[]{lll}\Delta_{A}\Delta_{B}&\geq|\langle A\rangle\langle
B\rangle-\langle AB\rangle|\\\
&=\sqrt{\frac{1}{4}|\langle[A,B]\rangle|^{2}+|\frac{1}{2}\langle\\{A,B\\}\rangle-\langle
A\rangle\langle B\rangle|^{2}}\\\
&\geq\frac{1}{2}|\langle[A,B]\rangle|.\end{array}$

Now we are at a position to show that Theorem 2.2 is sharper than the
uncertainty relations obtained by the approach mentioned in the introduction
section.

Let $A_{1},A_{2},\ldots,A_{k}$ be observables.

If $k=2n$ is even, by inequalities (3.2) one has

$None$
$\begin{array}[]{rl}\Pi_{j=1}^{k}\Delta_{A_{j}}=&(\Pi_{j=1}^{n-1}(\Delta_{A_{2j}}\Delta_{A_{2j+1}}))(\Delta_{A_{1}}\Delta_{A_{2n}})\\\
\geq&(\Pi_{j=1}^{n-1}|\langle A_{2j}A_{2j+1}\rangle-\langle
A_{2j}\rangle\langle A_{2j+1}\rangle|)\langle A_{1}A_{2n}\rangle-\langle
A_{1}\rangle\langle A_{2n}\rangle|\\\
\geq&\frac{1}{2^{n}}(\Pi_{j=1}^{n-1}|\langle[A_{2j},A_{2j+1}]\rangle|)|\langle[A_{1},A_{2n}]\rangle|,\end{array}$

which is weaker than the inequality (2.4) since
$\Delta_{A_{1}}\Delta_{A_{2n}}\geq|\langle A_{1}A_{2n}\rangle-\langle
A_{1}\rangle\langle A_{2n}\rangle|$.

If $k=2n+1$ is odd, by (3.2) again we have

$None$
$\begin{array}[]{rl}\Pi_{j=1}^{k}\Delta_{A_{j}}^{2}=&(\Pi_{j=1}^{n}(\Delta_{A_{2j-1}}\Delta_{A_{2j}}))(\Pi_{j=1}^{n}(\Delta_{A_{2}j}\Delta_{A_{2j+1}}))(\Delta_{A_{1}}\Delta_{A_{2n+1}})\\\
\geq&(\Pi_{j=1}^{n}|(\langle A_{2j-1}A_{2j}\rangle-\langle
A_{2j-1}\rangle\langle A_{2j}\rangle)(\langle A_{2j}A_{2j+1}\rangle-\langle
A_{2j}\rangle\langle A_{2j+1}\rangle)|)\cdot\\\ &\cdot|\langle
A_{1}A_{2n+1}\rangle-\langle A_{1}\rangle\langle A_{2n+1}\rangle|\\\
\geq&\frac{1}{2^{2(2n+1)}}(\Pi_{j=1}^{n}(|\langle[A_{2j-1},A_{2j}]\rangle\langle[A_{2j},A_{2j+1}]\rangle|))|\langle[A_{1},A_{2n+1}]\rangle|,\end{array}$

which is clearly weaker than the inequality (2.5) as $a^{2}+b^{2}\geq 2ab$ and
$\Delta_{A_{1}}\Delta_{A_{2n+1}}\geq|\langle A_{1}A_{2n+1}\rangle-\langle
A_{1}\rangle\langle A_{2n+1}\rangle|$.

##  4\. Uncertainty relations for three or four observables

By Theorem 2.2 and a careful check of its proof, one gets a uncertainty
relation for any three observables like the following.

Theorem 4.1. Let $A,B,C$ be three observables for a state $\rho$ in a state
space $H$, then

$None$
$\begin{array}[]{rl}\Delta_{A}^{2}\Delta_{B}^{2}\Delta_{C}^{2}\geq&\frac{1}{4}(\Delta_{C}^{2}|\langle
AB\rangle-\langle A\rangle\langle B\rangle|^{2}+\Delta_{A}^{2}|\langle
BC\rangle-\langle B\rangle\langle C\rangle|^{2})\\\ &+\frac{1}{2}|(\langle
AB\rangle-\langle A\rangle\langle B\rangle)(\langle BC\rangle-\langle
B\rangle\langle C\rangle)(\langle AC\rangle-\langle A\rangle\langle
C\rangle)|.\end{array}$

Particularly, for the case when $\Delta_{A}\Delta_{C}=|\langle
AC\rangle-\langle A\rangle\langle C\rangle|$ or $\langle AB\rangle=\langle
A\rangle\langle B\rangle$ or $\dim H=2$,

$None$ $\Delta_{A}\Delta_{B}\Delta_{C}\geq\frac{1}{2}(\Delta_{A}|\langle
BC\rangle-\langle B\rangle\langle C\rangle|+\Delta_{C}|\langle
AB\rangle-\langle A\rangle\langle B\rangle|).$

The inequalities (4.1) and (4.2) are tight as illustrated by applying to Pauli
matrices.

Example 4.2. Uncertainty relations for Pauli matrices.

Let $X,Y,Z$ be Pauli matrices, that is,

$X=\begin{pmatrix}0&1\cr 1&0\cr\end{pmatrix},\qquad Y=\begin{pmatrix}0&-i\cr
i&0\cr\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\cr 0&-1\cr\end{pmatrix}.$

Recall that, for any dense matrix $\rho\in M_{2}({\mathbb{C}})$, $\rho$ has a
representation

$\rho=\frac{1}{2}(I_{2}+r_{1}X+r_{2}Y+r_{3}Z)$

with Bloch vector $(r_{1},r_{2},r_{3})^{t}\in{\mathbb{R}}^{3}$ and
$r_{1}^{2}+r_{2}^{2}+r_{3}^{2}\leq 1$; and $\rho$ is pure if and only if
$r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=1$. Recall also that $XY=iZ$, $YZ=iX$ and
$\Delta_{A}^{2}=1-\langle A\rangle^{2}$ for $A\in\\{X,Y,Z\\}$, $\langle
X^{2}\rangle=\langle Y^{2}\rangle=\langle Z^{2}\rangle=1$ and $(\langle
X\rangle,\langle Y\rangle,\langle Z\rangle)=(r_{1},r_{2},r_{3})$.

Applying the inequality (4.2) of Theorem 4.1 to $X,Y,Z$ we get

$None$ $\begin{array}[]{rl}&\Delta_{X}\Delta_{Y}\Delta_{Z}\\\
\geq&\frac{1}{2}(\Delta_{X}|i\langle X\rangle-\langle Y\rangle\langle
Z\rangle|+\Delta_{Z}|i\langle Z\rangle-\langle X\rangle\langle Y\rangle|)\\\
=&\frac{1}{2}(\sqrt{(1-\langle X\rangle^{2})(\langle X\rangle^{2}+\langle
Y\rangle^{2}\langle Z\rangle^{2})}+\sqrt{(1-\langle Z\rangle^{2})(\langle
Z\rangle^{2}+\langle X\rangle^{2}\langle Y\rangle^{2})}\ ).\end{array}$

Obviously, the inequality (4.3) is tight and “=” holds if the Bloch vector
satisfies $|r_{1}|=|r_{3}|=\frac{1}{\sqrt{2}}$ and $r_{2}=0$.

This illustrates that Theorem 2.2 is tight for three observables. Since
Schrödinger’s uncertainty principle (1.3) is tight and our uncertainty
relation is equivalent to Schrödinger’s uncertainty principle by (3.2),
Theorem 2.2 is also tight for two obserables.

Moreover, by Theorem 3.1,

$(1-\langle Z\rangle^{2})(1-\langle X\rangle^{2})\geq\langle
Y\rangle^{2}+\langle X\rangle^{2}\langle Z\rangle^{2},$

hence we have

$None$ $\begin{array}[]{rl}&\Delta_{X}^{2}\Delta_{Y}^{2}\Delta_{Z}^{2}\\\
\geq&\frac{1}{4}[(1-\langle X\rangle^{2})(\langle X\rangle^{2}+\langle
Y\rangle^{2}\langle Z\rangle^{2})+(1-\langle Z\rangle^{2})(\langle
Z\rangle^{2}+\langle X\rangle^{2}\langle Y\rangle^{2})]\\\
&+\frac{1}{2}\sqrt{(1-\langle Z\rangle^{2})(1-\langle X\rangle^{2})(\langle
X\rangle^{2}+\langle Y\rangle^{2}\langle Z\rangle^{2})(\langle
Z\rangle^{2}+\langle X\rangle^{2}\langle Y\rangle^{2})}\\\
\geq&\frac{1}{4}[(1-\langle X\rangle^{2})(\langle X\rangle^{2}+\langle
Y\rangle^{2}\langle Z\rangle^{2})+(1-\langle Z\rangle^{2})(\langle
Z\rangle^{2}+\langle X\rangle^{2}\langle Y\rangle^{2})]\\\
&+\frac{1}{2}\sqrt{(\langle Y\rangle^{2}+\langle X\rangle^{2}\langle
Z\rangle^{2})(\langle X\rangle^{2}+\langle Y\rangle^{2}\langle
Z\rangle^{2})(\langle Z\rangle^{2}+\langle X\rangle^{2}\langle
Y\rangle^{2})}\\\ \geq&\sqrt{(\langle Y\rangle^{2}+\langle X\rangle^{2}\langle
Z\rangle^{2})(\langle X\rangle^{2}+\langle Y\rangle^{2}\langle
Z\rangle^{2})(\langle Z\rangle^{2}+\langle X\rangle^{2}\langle
Y\rangle^{2})}\\\ \geq&2\sqrt{2}|\langle X\rangle\langle Y\rangle\langle
Z\rangle|^{\frac{3}{2}}.\end{array}$

Particularly, one has

$None$ $\Delta_{X}^{2}\Delta_{Y}^{2}\Delta_{Z}^{2}\geq 2\sqrt{2}|\langle
X\rangle\langle Y\rangle\langle Z\rangle|^{\frac{3}{2}}.$

Observe that we always have

$1\geq\Delta_{X}^{2}\Delta_{Y}^{2}\Delta_{Z}^{2}=(1-r_{1}^{2})(1-r_{2}^{2})(1-r_{3}^{2})\geq\frac{8}{27}$

since the function $(1-r_{1}^{2})(1-r_{2}^{2})(1-r_{3}^{2})$ has its minimum
value $\frac{8}{27}$ at $|r_{1}|=|r_{2}|=|r_{3}|=\frac{1}{\sqrt{3}}$ and the
maximum value 1 at $\rho=\frac{1}{2}I_{2}$. Moreover, $|r_{1}r_{2}r_{3}|$
achieves simultaneously its maximum value $\frac{1}{3\sqrt{3}}$ at
$|r_{1}|=|r_{2}|=|r_{3}|=\frac{1}{\sqrt{3}}$. Thus the inequality (4.5) can be
sharped to

$None$
$\Delta_{X}^{2}\Delta_{Y}^{2}\Delta_{Z}^{2}\geq\frac{8\sqrt[4]{3}}{3}|\langle
X\rangle\langle Y\rangle\langle Z\rangle|^{\frac{3}{2}}.$

The inequality (4.6) is tight in the sense that “=” holds if
$|r_{1}|=|r_{2}|=r_{3}|=\frac{1}{\sqrt{3}}$.

Compare (4.3) with (1.6) and (4.6). Although these inequalities are all tight,
one of the remarkable advantage of (4.3) is that, even if some of $\langle
X\rangle,\langle Y\rangle,\langle Z\rangle$ are zero, we still may get a
positive lower bound of $\Delta_{X}\Delta_{Y}\Delta_{Z}$. For instance, saying
$\langle Y\rangle=0$, we have

$\Delta_{X}\Delta_{Y}\Delta_{Z}\geq\frac{1}{2}\left(\sqrt{(1-\langle
X\rangle^{2})\langle X\rangle^{2}}+\sqrt{(1-\langle Z\rangle^{2})\langle
Z\rangle^{2}}\right);$

saying $\langle Y\rangle=\langle Z\rangle=0$, we have

$\Delta_{X}\Delta_{Y}\Delta_{Z}\geq\frac{1}{2}\sqrt{(1-\langle
X\rangle^{2})\langle X\rangle^{2}},$

while we cannot get any information from (1.6) and (4.6).

Before conclusion we state the uncertainty relation from Theorem 2.2 for four
observations, which has a relatively simple expression.

Theorem 4.3. Let $A_{1},A_{2},A_{3},A_{4}$ be observables. Then

$None$
$\begin{array}[]{rl}&\Delta_{A_{1}}\Delta_{A_{2}}\Delta_{A_{3}}\Delta_{A_{4}}\\\
\geq&\frac{1}{2}|\langle A_{2}A_{3}\rangle-\langle A_{2}\rangle\langle
A_{3}\rangle|(|\langle A_{1}A_{4}\rangle-\langle A_{1}\rangle\langle
A_{4}\rangle|+\Delta_{A_{1}}\Delta_{A_{4}}).\end{array}$

The inequality (4.7) is tight. For example, Consider bipartite continuous-
variable system. Let
$(A_{1},A_{4},A_{2},A_{3})=(\hat{q}_{1},\hat{p}_{1},\hat{q}_{2},\hat{p}_{2})$,
where $\hat{q}_{i},\hat{p}_{i}$ are the position and momentum in the $i$th
mode satisfying the canonical commutation relation. As Heisenberg’s
uncertainty principle (1.1) is tight, we say that

$None$
$\begin{array}[]{rl}&\Delta_{{q}_{1}}\Delta_{{p}_{1}}\Delta_{{q}_{2}}\Delta_{{p}_{2}}\\\
\geq&\frac{1}{2}|\langle\hat{q}_{2}\hat{p}_{2}\rangle-\langle\hat{q}_{2}\rangle\langle\hat{p}_{2}\rangle|(|\langle\hat{q}_{1}\hat{p}_{1}\rangle-\langle\hat{q}_{1}\rangle\langle\hat{p}_{1}\rangle|+\Delta_{q_{1}}\Delta_{p_{4}}).\end{array}$

is tight, the “$=$” is attained at $\rho=e$.

Similarly, considering the positions and momentums
$(\hat{q}_{1},\hat{p}_{1},\hat{q}_{2},\hat{p}_{2},\ldots,\hat{q}_{n},\hat{p}_{n})$
in a $n$-partite continuous-variable system, one sees that the uncertainty
relation (2.4) in Theorem 2.2 is tight.

However we do not know whether the uncertainty relation (2.5) is tight for odd
$k=2n+1\geq 5$.

##  5\. Conclusion

Uncertainty relations discover lower bounds of the product of standard
deviations of several observables. Larger the lower bound is, more powerful
the corresponding uncertainty relation is. There are no known uncertainty
relations that valid for arbitrary multi observavles. By finding the equality
of deviation and the norm of the Lie product of the observable and the pure
state, we reduce the question of establishing uncertainty relation of multi
observerbles to the question of computing the numerical radius of an operator
of rank $\leq 2$. This enable us establish a universal uncertainty relation
for any $k$ observables, of which, the formulation depends on the even or odd
quality of $k$. For two observables, our uncertainty relation is exactly a
simpler reformulation of Schrödinger’s uncertainty principle. The uncertainty
relation provided in this paper is tight, at least for the cases of two and
three observables, as illustrated by examples.

## References

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##  6\. Appendix

In the appendix, we give the proofs of theorems 2.1 and 2.2.

###### Proof of Theorem 2.1.

Let $H$ be the associated Hilbert space for the pure state $|x\rangle$ and the
observable $A$. Write $A|x\rangle$ in the form
$A|x\rangle=\alpha|x\rangle+\beta|y\rangle$, where normalized $|y\rangle$ is
orthogonal to $|x\rangle$. Since $A$ is self-adjoint we have $\alpha=\langle
x|A|x\rangle\in\mathbb{R}$. Moreover, by self-adjointness of $A$, the Lie
product of $A$ and the rank one projection $|x\rangle\langle x|$ is
represented by the following matrix relative to decomposition
$H=[x]\oplus[y]\oplus\\{x,y,\\}^{\perp}$, here $[x]={\rm span}\\{x\\}$. Then

$[A,|x\rangle\langle x|]=\left(\begin{matrix}0&-\bar{\beta}\\\
\beta&0\end{matrix}\right)\oplus 0.$

Note that $[A,|x\rangle\langle x|]$ is a skew self-adjoint operator because
$[A,|x\rangle\langle x|]^{\dagger}=-[A,|x\rangle\langle x|]$. Thus its
numerical range $W([A,|x\rangle\langle x|])=i[-|\beta|,|\beta|]$, and hence
$w([A,|x\rangle\langle x|])=\|[A,x\otimes x]\|=|\beta|$. It follows that

$\begin{array}[]{lllll}w([A,|x\rangle\langle x|])^{2}&=\|[A,|x\rangle\langle
x|]\|=|\beta|^{2}\\\ &=\|A|x\rangle-\langle x|A|x\rangle\,|x\rangle\|^{2}\\\
&=\langle x|(A-\langle x|A|x\rangle)^{2}|x\rangle\\\ &=\langle
x|A^{2}|x\rangle-(\langle x|A|x\rangle)^{2}\\\ &=\langle A^{2}\rangle-\langle
A\rangle^{2}=\Delta_{A}^{2}.\end{array}$

Therefore, $\Delta_{A}=w([A,|x\rangle\langle x|])=\|[A,|x\rangle\langle
x|]\|$, completing the proof. ∎

Before start the proof of Theorem 2.2, we need a lemma.

Lemma A1. Let

$E_{1}=\left(\begin{array}[]{ccc}0&a&b\\\ c&0&0\\\
0&0&0\end{array}\right),E_{2}=\left(\begin{array}[]{cc}0&a\\\
c&0\end{array}\right).$

Then

$w(E_{1})=\frac{1}{2}\sqrt{|b|^{2}+(|a|+|c|)^{2}}$

and

$w(E_{2})=\frac{1}{2}(|a|+|c|).$

###### Proof.

Obviously, with $ac=|ac|e^{2i\theta}$,
$\sigma(E_{1})=\\{\pm\sqrt{|ac|}e^{i\theta},0\\}$. It is easily checked that
$E_{1}$ satisfies the conditions of Theorem 2.3 and 2.4 of Ref [14], and hence
the numerical range $W(E_{1})$ of $E_{1}$ is an elliptic disc with foci
$\\{\pm\sqrt{|ac|}e^{i\theta}\\}$. Thus the numerical radius $w(E_{1})$ is the
half length of major axis of the ellipse.

Let $F=e^{-i\theta}E_{1}$; then $w(F)=w(E_{1})$. As
$\sigma(F)=\\{\pm\sqrt{|ac|},0\\}$, we see that

$w(F)=\|{\rm Re}(F)\|.$

Note that

$\sigma({\rm
Re}(F))=\\{0,\pm\frac{1}{2}\sqrt{|b|^{2}+|ae^{-i\theta}+\bar{c}e^{i\theta}|^{2}}\
\\}.$

A simple computation shows that

$|ae^{-i\theta}+\bar{c}e^{i\theta}|^{2}=(|a|+|c|)^{2}.$

Therefore, we have

$w(E_{1})=w(F)=\frac{1}{2}\sqrt{|b|^{2}+(|a|+|c|)^{2}}.$

It is also easily checked that

$w(E_{2})=\frac{1}{2}|ae^{-i\theta}+\bar{c}e^{i\theta}|=\frac{1}{2}(|a|+|c|).$

∎

###### Proof of Theorem 2.2.

As remarked after the statement os Theorem 2.2, it is enough to prove the
theorem for the pure states.

For any given observables $A_{1},A_{2},\ldots,A_{k}$ with $k\geq 2$, let
$D_{k}=\Pi_{j=1}^{k}[A_{j},|x\rangle\langle x|]$, which has the form

$D_{k}=a_{k}|x\rangle\langle x|+b_{k}|A_{1}x\rangle\langle
x|+c_{k}|x\rangle\langle xA_{k}|+d_{k}|A_{1}x\rangle\langle xA_{k}|.$

A direct computation gives

$None$ $\left\\{\begin{array}[]{l}a_{2}=-\langle A_{1}A_{2}\rangle,\\\
b_{2}=\langle A_{2}\rangle,\\\ c_{2}=\langle A_{1}\rangle,\\\
d_{2}=-1.\end{array}\right.$

For any $k\geq 3$, since $D_{k}=D_{k-1}[A_{k},|x\rangle\langle x|]$, one may
take

$None$ $\left\\{\begin{array}[]{l}a_{k}=a_{k-1}\langle
A_{k}\rangle+c_{k-1}\langle A_{k-1}A_{k}\rangle,\\\ b_{k}=b_{k-1}\langle
A_{k}\rangle+d_{k-1}\langle A_{k-1}A_{k}\rangle,\\\
c_{k}=-a_{k-1}-c_{k-1}\langle A_{k-1}\rangle,\\\ d_{k}=-b_{k-1}-d_{k-1}\langle
A_{k-1}\rangle.\end{array}\right.$

Take unitors vectors $|y\rangle,|z\rangle$ so that
$\\{|x\rangle,|y\rangle,|z\rangle\\}$ is orthogonal and

$\left\\{\begin{array}[]{l}|A_{1}x\rangle=\langle
A_{1}\rangle|x\rangle+\Delta_{A_{1}}|y\rangle,\\\ |A_{k}x\rangle=\langle
A_{k}\rangle|x\rangle+\beta^{\prime}|y\rangle+\gamma^{\prime}|z\rangle.\end{array}\right.$

Then

$None$ $\beta^{\prime}=\Delta_{A_{1}}^{-1}(\langle A_{1}A_{k}\rangle-\langle
A_{1}\rangle\langle
A_{k}\rangle),\quad\Delta_{A_{k}}=\sqrt{|\beta^{\prime}|^{2}+|\gamma^{\prime}|^{2}}.$

and

$None$ $\begin{array}[]{rl}D_{k}=&(a_{k}+b_{k}\langle
A_{1}\rangle+c_{k}\langle A_{k}\rangle+d_{k}\langle A_{1}\rangle\langle
A_{k}\rangle)|x\rangle\langle x|+(c_{k}+d_{k}\langle
A_{1}\rangle)\bar{\beta^{\prime}}|x\rangle\langle y|\\\ &+(c_{k}+d_{k}\langle
A_{1}\rangle)\bar{\gamma^{\prime}}|x\rangle\langle z|+(b_{k}+d_{k}\langle
A_{k}\rangle)\Delta_{A_{1}}|y\rangle\langle x|\\\
&+d_{k}\Delta_{A_{1}}\bar{\beta^{\prime}}|y\rangle\langle
y|+d_{k}\Delta_{A_{1}}\bar{\gamma^{\prime}}|y\rangle\langle z|\\\
=&f_{11}^{(k)}|x\rangle\langle x|+f_{12}^{(k)}|x\rangle\langle
y|+f_{13}^{(k)}|x\rangle\langle z|\\\ &+f_{21}^{(k)}|y\rangle\langle
x|+f_{22}^{(k)}|y\rangle\langle y|+f_{23}^{(k)}|y\rangle\langle
z|.\end{array}$

Note that, by (A2), we have

$\begin{array}[]{rl}f_{11}^{(k)}=&c_{k-1}(\langle A_{k-1}A_{k}\rangle-\langle
A_{k-1}\rangle\langle A_{k}\rangle)+d_{k-1}(\langle A_{1}\rangle\langle
A_{k-1}A_{k}\rangle-\langle A_{1}\rangle\langle A_{k-1}\rangle\langle
A_{k}\rangle)\\\ =&-f_{11}^{(k-2)}(\langle A_{k-1}A_{k}\rangle-\langle
A_{k-1}\rangle\langle A_{k}\rangle)\end{array}$

as

$\begin{array}[]{rl}&c_{k}+d_{k}\langle A_{1}\rangle\\\ =&-c_{k-2}(\langle
A_{k-2}A_{k-1}\rangle-\langle A_{k-2}\rangle\langle
A_{k-1}\rangle)-d_{k-2}(\langle A_{1}\rangle\langle
A_{k-2}A_{k-1}\rangle-\langle A_{1}\rangle\langle A_{k-2}\rangle\langle
A_{k-1}\rangle)\\\ =&-f_{11}^{(k-1)},\end{array}$
$d_{k}=-b_{k-1}-d_{k-1}\langle A_{k-1}\rangle=d_{k-2}(\langle
A_{k-2}\rangle\langle A_{k-1}\rangle-\langle A_{k-2}A_{k-1}\rangle)$

and

$b_{k}+d_{k}\langle A_{k}\rangle=d_{k-1}(\langle A_{k-1}A_{k}\rangle-\langle
A_{k-1}\rangle\langle A_{k}\rangle)=-d_{k+1},$

which reveal that

$None$ $\begin{array}[]{l}f_{11}^{(k)}=-f_{11}^{(k-2)}(\langle
A_{k-1}A_{k}\rangle-\langle A_{k-1}\rangle\langle A_{k}\rangle),\\\
f_{12}^{(k)}=-f_{11}^{(k-1)}\bar{\beta^{\prime}},\\\
f_{13}^{(k)}=-f_{11}^{(k-1)}\bar{\gamma^{\prime}},\\\
f_{21}^{(k)}=-d_{k+1}\Delta_{A_{1}}=(\langle A_{k-1}A_{k}\rangle-\langle
A_{k-1}\rangle\langle A_{k}\rangle)f_{21}^{(k-2)},\\\
f_{22}^{(k)}=d_{k}\Delta_{A_{1}}\bar{\beta^{\prime}}=(\langle
A_{k-2}A_{k-1}\rangle-\langle A_{k-2}\rangle\langle
A_{k-1}\rangle)d_{k-2}\Delta_{A_{1}}\bar{\beta^{\prime}},\\\
f_{23}^{(k)}=d_{k}\Delta_{A_{1}}\bar{\gamma^{\prime}}=(\langle
A_{k-2}A_{k-1}\rangle-\langle A_{k-2}\rangle\langle
A_{k-1}\rangle)d_{k-2}\Delta_{A_{1}}\bar{\gamma^{\prime}}.\end{array}$

It is easily checked that

$D_{2}=\begin{array}[]{ccc}&\left(\begin{matrix}\langle A_{1}\rangle\langle
A_{2}\rangle-\langle A_{1}A_{2}\rangle&0&0\\\
0&-\Delta_{A_{1}}\beta^{\prime}&-\Delta_{A_{1}}\gamma^{\prime}\\\ 0&0&0\\\
\end{matrix}\right)\oplus 0\end{array}$

and

$D_{3}=\left(\begin{array}[]{ccc}0&(\langle A_{1}A_{2}\rangle-\langle
A_{1}\rangle\langle A_{2}\rangle)\beta^{\prime}&(\langle
A_{1}A_{2}\rangle-\langle A_{1}\rangle\langle A_{2}\rangle)\gamma^{\prime}\\\
(\langle A_{2}\rangle\langle A_{3}\rangle-\langle
A_{2}A_{3}\rangle)\Delta_{A_{1}}&0&0\\\ 0&0&0\end{array}\right)\oplus 0$

if $\dim H\geq 3$;

$D_{2}=\left(\begin{array}[]{cc}\langle A_{1}\rangle\langle
A_{2}\rangle-\langle A_{1}A_{2}\rangle&0\\\
0&-\Delta_{A_{1}}\Delta_{A_{2}}\end{array}\right)$

and

$D_{3}=\left(\begin{array}[]{cc}0&(\langle A_{1}A_{2}\rangle-\langle
A_{1}\rangle\langle A_{2}\rangle)\Delta_{A_{2}}\\\ (\langle
A_{2}\rangle\langle A_{3}\rangle-\langle
A_{2}A_{3}\rangle)\Delta_{A_{1}}&0\end{array}\right)$

if $\dim H=2$. Hence, by identify the case of $\dim H=2$ with the case
$\gamma^{\prime}=0$, we may agree that $D_{2}$ and $D_{3}$ have respectively
the patten

$None$ $\left(\begin{array}[]{ccc}*&0&0\\\ 0&*&*\\\
0&0&0\end{array}\right)\oplus 0$

and

$None$ $\left(\begin{array}[]{ccc}0&*&*\\\ &0&0\\\
0&0&0\end{array}\right)\oplus 0.$

Then Eq.(A5) implies that $D_{2n}$ has the patten (A6) if $k=2n$ is even and
$D_{2n+1}$ has the patten (A7)) if $k=2n+1$ is odd.

Let us first calculate $w(D_{2})$. It is easily checked that

$\Delta_{A_{1}}\Delta_{A_{2}}\geq w(D_{2})=\max\\{|\langle
A_{1}A_{2}\rangle-\langle A_{1}\rangle\langle
A_{2}\rangle|,\frac{1}{2}|\langle A_{1}A_{2}\rangle-\langle
A_{1}\rangle\langle A_{2}\rangle|+\frac{\Delta_{A_{1}}\Delta_{A_{2}}}{2}\\}.$

Thus we have

$None$ $\Delta_{A_{1}}\Delta_{A_{2}}\geq|\langle A\rangle\langle
B\rangle-\langle AB\rangle|$

and

$w(D_{2})=\frac{1}{2}(|\langle A_{1}A_{2}\rangle-\langle A_{1}\rangle\langle
A_{2}\rangle|+{\Delta_{A_{1}}\Delta_{A_{2}}}).$

If $k=2n$ is even, then it follows from (A5) that

$None$ $|f_{11}^{(2n)}|=|(\langle A_{1}A_{2}\rangle-\langle
A_{1}\rangle\langle A_{1}\rangle)\cdots(\langle A_{k-1}A_{k}\rangle-\langle
A_{k-1}\rangle\langle A_{k}\rangle)|.$

Since $D_{2n}$ has patten (A6), by (A5) and (A8), we have

$\|D_{2n}\|=\max\\{|f_{11}^{(2n)}|,\sqrt{|f_{22}^{(2n)}|^{2}+|f_{23}^{(2n)}|^{2}}\
\\}=|d_{2n}|\Delta_{A_{1}}\Delta_{A_{2n}}$

and

$\begin{array}[]{rl}w(D_{2n})=&\max\\{|f_{11}^{(2n)}|,\frac{1}{2}\left(|f_{22}^{(2n)}|+\sqrt{|f_{22}^{(2n)}|^{2}+|f_{23}^{(2n)}|^{2}}\right)\\}\\\
=&\frac{1}{2}|d_{2n}|(|\langle A_{1}A_{2n}\rangle-\langle A_{1}\rangle\langle
A_{2n}\rangle|+\Delta_{A_{1}}\Delta_{A_{2n}}).\end{array}$

Now $d_{2}=-1$ and

$|d_{2n}|=|d_{2n-2}|\cdot|\langle A_{2n-2}A_{2n-1}\rangle-\langle
A_{2n-2}\rangle\langle A_{2n-1}\rangle|$

entails that

$None$ $\begin{array}[]{rl}w(D_{2n})=\frac{1}{2}(\Pi_{j=1}^{n-1}|\langle
A_{2j}A_{2j+1}\rangle-\langle A_{2j}\rangle\langle A_{2j+1}\rangle|)(|\langle
A_{1}A_{2n}\rangle-\langle A_{1}\rangle\langle
A_{2n}\rangle|+\Delta_{A_{1}}\Delta_{A_{2n}}).\end{array}$

As $\Pi_{j=1}^{2n}\Delta_{A_{j}}\geq w(D_{2n})$, this completes the proof of
(2.4).

If $k=2n+1$ is odd, then $D_{2n+1}$ has the patten (A7). Applying Lemma A1
gives

$\begin{array}[]{rl}w(D_{2n+1})=&\frac{1}{2}\sqrt{(|f_{12}^{(2n+1)}|+|f_{21}^{(2n+1)}|)^{2}+|f_{13}^{(2n+1)}|^{2}}\\\
=&\frac{1}{2}\sqrt{(|f_{11}^{(2n)}\beta^{\prime}|+|d_{(2n+2)}|\Delta_{A_{1}})^{2}+|f_{11}^{(2n)}\gamma^{\prime}|^{2}}\\\
=&\frac{1}{2}\sqrt{(2|f_{11}^{(2n)}d_{(2n+2)}|\Delta_{A_{1}}\beta^{\prime}|+|d_{(2n+2)}|^{2}\Delta_{A_{1}}^{2}+|f_{11}^{(2n)}|^{2}(|\beta^{\prime}|^{2}+|\gamma^{\prime}|^{2})}.\end{array}$

Therefore,

$None$ $w(D_{2n+1})=\frac{1}{2}\sqrt{2\pi_{1}\pi_{2}|\langle
A_{1}A_{2n+1}\rangle-\langle A_{1}\rangle\langle
A_{2n+1}\rangle|+\pi_{2}^{2}\Delta_{A_{1}}^{2}+\pi_{1}^{2}\Delta_{A_{2n+1}}^{2}},$

where

$\pi_{1}=\Pi_{j=1}^{n}|\langle A_{2j-1}A_{2j}\rangle-\langle
A_{2j-1}\rangle\langle A_{2j}\rangle|$

and

$\pi_{2}=\Pi_{j=1}^{n}|\langle A_{2j}A_{2j+1}\rangle-\langle
A_{2j}\rangle\langle A_{2j+1}\rangle|.$

As $\Pi_{j=1}^{2n+1}\Delta_{A_{j}}\geq w(D_{2n+1})$, we complete the proof of
(2.5) by (A11). ∎

Finally we explain why the sharper inequality

$None$ $\Pi_{j=1}^{k}\Delta_{A_{j}}\geq\|D_{k}\|$

cannot achieve sharper uncertainty relations than the weaker inequality
$\Pi_{j=1}^{k}\Delta_{A_{j}}\geq w(D_{k})$ can.

If $k=2n$ is even, then by (A5) and (A6) one has

$\|D_{2n}\|=\Delta_{A_{1}}\Delta_{A_{2n}}\Pi_{j=1}^{n-1}|\langle
A_{2j}A_{2j+1}\rangle-\langle A_{2j}\rangle\langle A_{2j+1}\rangle|.$

This together with (A12) gives

$\Pi_{j=2}^{2n-1}\Delta_{A_{j}}\geq\Pi_{j=1}^{n-1}|\langle
A_{2j}A_{2j+1}\rangle-\langle A_{2j}\rangle\langle A_{2j+1}\rangle|,$

which is weaker than the inequality (2.4).

If $k=2n+1$ is odd, then by (A5) and (A7) one has

$\begin{array}[]{rl}\|D_{2n+1}\|=&\max\\{\sqrt{|f_{12}^{(2n+1)}|^{2}+|f_{13}^{(2n+1)}|^{2}},|f_{21}^{(2n+1)}|\\}\\\
=&\max\\{|f_{11}^{(2n)}|\Delta_{A_{2n+1}},|d_{2n+2}|\Delta_{A_{1}}\\},\end{array}$

which gives

$\Pi_{j=1}^{2n}\Delta_{A_{j}}\geq\Pi_{j=1}^{n}|\langle
A_{2j-1}A_{2j}\rangle-\langle A_{2j-1}\rangle\langle A_{2j}\rangle|$

or

$\Pi_{j=2}^{2n+1}\Delta_{A_{j}}\geq\Pi_{j=1}^{n}|\langle
A_{2j}A_{2j+1}\rangle-\langle A_{2j}\rangle\langle A_{2j+1}\rangle|,$

again weaker than (2.4).

Acknowledgements This work is partly supported by National Science Foundation
of China (11201329, 11171249) and Program for the Outstanding Innovative Teams
of Higher Learning Institutions of Shanxi.

Competing interests statement The authors declare that they have no competing
financial interests.

Correspondence should be addressed to J. C. Hou (houjinchuan@tyut.edu.cn).