# Discrimination of POVMs with rank-one effects

Aleksandra Krawiec1,∗ and Łukasz Pawela1 and Zbigniew Puchała1,2 1 Institute
of Theoretical and Applied Informatics, Polish Academy of Sciences, ul.
Bałtycka 5, 44-100 Gliwice, Poland  2 Faculty of Physics, Astronomy and
Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348
Kraków, Poland<EMAIL_ADDRESS>
###### Abstract.

The main goal of this work is to provide an insight into the problem of
discrimination of positive operator valued measures with rank-one effects. It
is our intention to study multiple shot discrimination of such measurements,
that is the case when we are able to use to unknown measurement a given number
of times. Furthermore, we are interested in comparing two possible
discrimination schemes: the parallel and adaptive ones. To this end we
construct a pair of symmetric, information complete positive operator valued
measures which can be perfectly discriminated in a two-shot adaptive scheme.
On top of this we provide an explicit algorithm which allows us to find this
adaptive scheme.

##  1\. Introduction

The transformation of quantum states is at the core of quantum computing. The
most general mathematical tool describing how quantum states transform are
quantum channels. While performing computation, one would like to certify that
the operation in use agrees with the one given by the classical description.
The discrimination task is closely related to the hypothesis testing. This is
one of the reasons why the problem of discrimination of quantum channels has
attracted a lot of attention [1, 2, 3, 4].

Various approaches have been utilized to study the distinguishability of
quantum channels. In work [5] the authors introduced the resource theory of
asymmetric distinguishability for quantum channels which was further developed
in [6]. A novel scenario called coherent quantum channel discrimination was
studied in [7]. Interesting results on the distinguishability concern specific
classes of quantum channels like unitary [8, 9] and Pauli [10] channels.

We can consider two approaches to channel discrimination – a parallel and an
adaptive scheme. In the former, we are allowed to perform the unknown channel
a fixed, finite, number of times on an input state, while in the latter case
we can also perform additional operations between the applications of the
unknown channel. In general it is a nontrivial question which discrimination
scheme should be used to distinguish between two quantum channels. It seems
natural that the use of adaptive scheme should improve the discrimination.
However, it may happen that it suffices to use the parallel one. For example
in the case of unitary channels [8, 9] and von Neumann measurements it was
shown in [11, 12, 13] that the parallel scheme is always optimal. It is also
known that asymptotically the use of adaptive strategy does not give an
advantage over the parallel one for the discrimination of classical [14] and
classical-quantum channels [15]. Nevertheless, an example of a pair of quantum
channels that cannot be distinguished perfectly in parallel but can be
distinguished by the adaptive scheme was proposed in [16]. These channels are
however pretty artificial. We will give another, natural example of two
channels which have this property. These channels are two symmetric
informationally complete positive operator valued measures (SIC POVMs) [17,
18] of dimension three. Moreover, we will show that there exist many pairs of
channels which can be distinguished only adaptively and present numerical
results for the discrimination of random symmetric operator valued measures
(POVMs).

In this work we focus on the following scheme. There are two measurements
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. One of them is secretly chosen (with
probability $\frac{1}{2}$) and put into a black box. The black box containing
the unknown measurement device can be used any finite number of times in any
configuration. Two schemes of multiple-shot discrimination will be studied:
parallel and adaptive. The parallel scheme does not require any processing
between the uses of the black box [19]. In the adaptive scheme, however, the
processing between the uses of the black box plays a crucial role. We can
prepare any input state and put it through the discrimination network.
Finally, we perform a known measurement and make a decision which measurement
was put into the black box.

This work is organized as follows. We begin with preliminaries in Section 2.
The conditions allowing for perfect discrimination in both parallel and
adaptive scenarios are stated in Section 3. The example of qutrit SIC POVMs
which can be discriminated perfectly only by the adaptive scheme, as well as
the algorithm allowing for perfect discrimination, are presented in Section 4.
Eventually, in Section 5 one can find numerical results for the discrimination
of random rank-one POVMs.

##  2\. Preliminaries

Let $\mathcal{X}$ and $\mathcal{Y}$ be finite-dimensional complex Hilbert
spaces. We denote by $\textrm{L}(\mathcal{X},\mathcal{Y})$ the set of linear
operators $A:\mathcal{X}\rightarrow\mathcal{Y}$ and write shortly
$\textrm{L}(\mathcal{X})$ for $\textrm{L}(\mathcal{X},\mathcal{X})$. We will
use the notation $\mathrm{U}(\mathcal{X},\mathcal{Y})$ for the set of isometry
operators and $\mathrm{U}(\mathcal{X})$ for the set of unitary operators. A
set of quantum states will be denoted $\mathcal{D}(\mathcal{X})$. Quantum
operations $\Phi:\textrm{L}(\mathcal{X})\rightarrow\textrm{L}(\mathcal{Y})$
will be denoted $\mathrm{T}(\mathcal{X},\mathcal{Y})$ while the subset of
quantum channels will be denoted $\mathrm{C}(\mathcal{X},\mathcal{Y})$.

We will need the notion of vectorization of a matrix. For the identity
operator ${\rm 1l}$ of the space $\mathrm{L}(\mathcal{X})$ we define its
vectorization as $|{\rm 1l}\rangle\\!\rangle=\sum_{i}|ii\rangle$. It can be
generalized to an arbitrary matrix $X\in\mathrm{L}(\mathcal{X})$ as
$|X\rangle\\!\rangle=(X\otimes{\rm 1l})|{\rm 1l}\rangle\\!\rangle$.

The most general form of quantum measurements are positive operator valued
measures (POVMs). Formally, a collection of positive semidefinite operators
$\mathcal{P}=\\{E_{1},\ldots,E_{m}\\}\subset\mathrm{L}(\mathcal{X})$ is called
a POVM iff $\sum_{i=1}^{m}E_{i}={\rm 1l}$. The operators $E_{i}$ are called
_effects_. Every quantum measurement $\mathcal{P}=\\{E_{1},\ldots,E_{m}\\}$
can be identified with a quantum channel which gives a classical output. Its
action on a quantum state $\rho$ can be written as

(1)
$\mathcal{P}(\rho)=\sum_{i=1}^{m}\operatorname{Tr}(E_{i}\rho)|i\rangle\langle
i|.$

Let us for a moment focus on the distinguishability of general quantum
channels. This problem has been widely studied in the literature in various
settings and utilizing a plethora of mathematical tools [20, 21]. The
probability of correct discrimination between channels
$\Phi,\Psi\in\mathrm{C}(\mathcal{X},\mathcal{Y})$ is upper-bounded by the
Holevo-Helstrom theorem [22]

(2) $p\leq\frac{1}{2}+\frac{1}{4}\|\Phi-\Psi\|_{\diamond}.$

The diamond norm, known also as the completely bounded trace norm,
$\|\cdot\|_{\diamond}$ is defined for a Hermiticity-preserving linear map
$\Phi\in\mathrm{T}(\mathcal{X},\mathcal{Y})$ as

(3) $\|\Phi\|_{\diamond}=\max_{\|X\|_{1}=1}\left\|\left(\Phi\otimes{\rm
1l}_{\mathrm{L}(\mathcal{X})}\right)(X)\right\|_{1}.$

where $\|X\|_{1}$ is defined as a sum of singular values of $X$. The above-
mentioned bound describes the situation when the black box containing the
unknown POVM can be used only once. Therefore $\Phi$ and $\Psi$ are said to be
perfectly distinguishable in a single-shot scenario iff
$\|\Phi-\Psi\|_{\diamond}=2$.

###  2.1. Parallel discrimination

The approach introduced by Holevo-Helstrom theorem can be easily generalized
into the parallel discrimination scheme. Is such a case the probability of
correct discrimination after $N$ uses of the black box is upper-bounded by

(4) $p\leq\frac{1}{2}+\frac{1}{4}\left\|\Phi^{\otimes N}-\Psi^{\otimes
N}\right\|_{\diamond}.$

The parallel discrimination scheme is depicted in Figure 1. Black boxes
containing the unknown POVM are denoted by question marks. As an input state
we use the state on $N+1$ registers denoted by $|\psi_{1,\ldots,N+1}\rangle$.
After performing the unknown POVM on $N$ registers we obtain classical outputs
$i_{1},\ldots,i_{N}$. Using this classical information we prepare another
known measurement and perform it on the last register. Basing on the last
measurement’s outcome we make a decision which measurement was hidden in the
black box.

Figure 1. Parallel discrimination scheme

We can state a condition when two measurements can be distinguished perfectly
in a parallel scenario. It follows directly from the Theorem 1 which will be
presented in the subsequent section. Let
$\mathcal{P}_{1}=\\{|x_{1}\rangle\langle x_{1}|,\ldots,|x_{m}\rangle\langle
x_{m}|\\}$ and $\mathcal{P}_{2}=\\{|y_{1}\rangle\langle
y_{1}|,\ldots,|y_{m}\rangle\langle y_{m}|\\}$ be POVMs for $m\geq d$. Then
POVMs $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ have perfect parallel
distinguishability if there exists an integer $N>0$ and a quantum state
$\rho\in\mathcal{D}(\mathcal{X}^{\otimes N})$ such that for all multi-indices
$i_{1},\ldots,i_{N}$ we have

(5) $\langle x_{i_{1}}\ldots x_{i_{N}}|\rho|y_{i_{1}}\ldots
y_{i_{N}}\rangle=0.$

###  2.2. Adaptive discrimination scheme

If we are allowed to perform processing between the uses of black boxes, then
we arrive at the adaptive discrimination scheme which is depicted in Figure 2.
Although the adaptive discrimination scheme can be used for any number of
uses, $N$, of the black box, for the sake of clarity let us focus on the case
$N=3$. We prepare an input state $|\psi\rangle$ and perform the unknown POVM
on the first register. Basing on the measurement’s outcome $i_{1}$ we perform
a quantum channel $\Phi_{i_{1}}$ on the other registers. Later, we perform the
unknown POVM on the second register and obtain the classical outcome $i_{2}$.
Then we perform a channel $\Phi_{i_{1},i_{2}}$ on the remaining registers
which now depends on both $i_{1}$ and $i_{2}$. In what follows we perform the
unknown POVM on the third register and obtain the outcome $i_{3}$. Finally, we
perform the known measurement on the last register which now depends on all
classical outcomes $i_{1},i_{2},i_{3}$ and make a decision which of the POVMs
was hidden in the black box.

Figure 2. Adaptive discrimination scheme

##  3\. Parallel vs adaptive discrimination scheme

In this section we will state the condition when a pair of quantum operations
can be distinguished perfectly in the adaptive scheme while the parallel
scheme does not allow for perfect discrimination. Before reviewing some known
results on the distinguishability of general quantum operations, we need to
introduce the notion of disjointness between quantum operations. Let
$\rho\in\mathcal{D}(\mathcal{X})$ be a quantum state with spectral
decomposition
$\rho=\sum_{i}\lambda_{i}|\lambda_{i}\rangle\langle\lambda_{i}|$. Then we
define its support as
$\textrm{supp}(\rho)=\textrm{span}\\{|\lambda_{i}\rangle:\lambda_{i}>0\\}$.

######  Definition 1.

Two quantum states $\rho_{1},\rho_{2}\in\mathcal{D}(\mathcal{X})$ are called
disjoint when

(6) $\textrm{supp}(\rho_{1})\cap\textrm{supp}(\rho_{2})=\\{0\\}.$

The notion of disjointness can be generalized to quantum operations. Thus,
following [23], we have the following definition.

######  Definition 2.

Two quantum operations $\Phi,\Psi\in\mathrm{T}(\mathcal{X},\mathcal{Y})$ are
called (entanglement-assisted) disjoint if there exists an input state
$|\psi\rangle\in\mathcal{X}\otimes\mathcal{Z}$ such that
$\left(\Phi\otimes{\rm
1l}_{\mathrm{L}(\mathcal{Z})}\right)(|\psi\rangle\langle\psi|)$ and
$\left(\Psi\otimes{\rm
1l}_{\mathrm{L}{(\mathcal{Z})}}\right)(|\psi\rangle\langle\psi|)$ are
disjoint.

Let $\Phi,\Psi\in\mathrm{T}(\mathcal{X},\mathcal{Y})$ be quantum operations
with Kraus operators $\\{E_{i}\\}_{i}$ and $\\{F_{i}\\}_{i}$ respectively.
From [23] we have a condition for perfect distinguishability by a finite
number of uses given by the following theorem.

######  Theorem 1.

Quantum operations $\Phi:\rho\mapsto\sum_{i}E_{i}\rho E_{i}^{\dagger}$ and
$\Psi:\rho\mapsto\sum_{j}F_{j}\rho F_{j}^{\dagger}$ are perfectly
distinguishable by a finite number of uses iff they are disjoint and ${\rm
1l}\not\in\mathrm{span}\\{E_{i}^{\dagger}F_{j}\\}_{i,j}$.

The following corollary comes from [19] and gives a condition when quantum
operations cannot be distinguished in parallel.

######  Corollary 1.

Using notation as in Theorem 1, if
$\mathrm{span}\\{E_{i}^{\dagger}F_{j}\\}_{i,j}$ contains a positive operator
$\rho>0$, then $\Phi$ and $\Psi$ cannot be distinguished perfectly in parallel
by any finite number of uses.

Combining the above we obtain the corollary which states a condition when
operations can be distinguished only by the adaptive scheme, that is when
parallel discrimination is not sufficient but adaptive discrimination allows
for perfect discrimination.

######  Corollary 2.

Using notation as in Theorem 1, if
$\mathrm{span}\\{E_{i}^{\dagger}F_{j}\\}_{i,j}$ contains a positive operator
$\rho>0$, operations $\Phi$ and $\Psi$ are (entanglement-assisted) disjoint
and ${\rm 1l}\not\in\mathrm{span}\\{E_{i}^{\dagger}F_{j}\\}_{i,j}$, then
$\Phi$ and $\Psi$ can be distinguished perfectly only by the adaptive scheme.

Let $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ be rank-one POVMs with effects
$\\{|x_{i}\rangle\langle x_{i}|\\}_{i}$ and $\\{|y_{i}\rangle\langle
y_{i}|\\}_{i}$ respectively. Therefore the Kraus operators of
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ can be chosen as
$E_{i}=\\{|i\rangle\langle x_{i}|\\}_{i}$ and $F_{i}=\\{|i\rangle\langle
y_{i}|\\}_{i}$ respectively. Hence in order to check whether ${\rm
1l}\notin\textrm{span}\\{E_{i}^{\dagger}F_{j}\\}_{i,j}$ it suffices to see
whether ${\rm 1l}\notin\textrm{span}\\{|x_{i}\rangle\langle y_{i}|\\}_{i}$ as
$E_{i}^{\dagger}F_{j}=|x_{i}\rangle\langle y_{j}|\delta_{ij}$.

##  4\. SIC POVMs

In this section we give an example of a pair of quantum measurements for which
the use of parallel discrimination scheme does not give perfect
distinguishability but they can be distinguished perfectly by the adaptive
scheme. Let us begin with a definition. A POVM
$\mathcal{P}=\\{E_{1},\ldots,E_{d^{2}}\\}\subset\mathrm{L}(\mathcal{X}_{d})$
of dimension $d$ is called a SIC POVM if
$E_{i}=\frac{1}{d}|\phi_{i}\rangle\langle\phi_{i}|$ and

(7) $\left|\langle{\phi_{i}}|{\phi_{j}}\rangle\right|^{2}=\frac{1}{d+1}$

for every $i,j=1,\ldots,d^{2}$, $i\neq j$. It is worth mentioning here that it
is an open question whether SIC POVMs exist in all dimensions. So far
numerical results are known for all dimensions up to $151$ and for a few other
dimensions up to $844$ [24].

In what follows we will construct an example of a pair of quantum channels
which cannot be distinguished perfectly in the parallel scheme but can be
distinguished perfectly in the adaptive scheme. Let $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ be qutrit SIC POVMs with effects $\\{|x_{i}\rangle\langle
x_{i}|\\}_{i}$, where $|x_{i}\rangle=\frac{1}{\sqrt{3}}|\phi_{i}\rangle$, and
$\\{|y_{i}\rangle\langle y_{i}|\\}_{i}$ respectively. Let
$\pi=(9,8,7,3,1,2,6,4,5)$ be a permutation such that
$|y_{i}\rangle=|x_{\pi(i)}\rangle$ for $i=1,\ldots,9$. A detailed construction
of the POVM $\\{|x_{i}\rangle\langle x_{i}|\\}_{i}$ can be found in [25] and
the exact form of the fiducial vector is stated in the supplementary material
of [26]. In this case ${\rm 1l}\not\in\mathrm{span}\\{|x_{i}\rangle\langle
y_{i}|\\}_{i}$ and
$\mathcal{P}_{1},\mathcal{P}_{2}\in\textrm{C}(\mathcal{X},\mathcal{Y})$ are
(entanglement-assisted) disjoint. To see the latter define two states

(8) $\begin{split}\rho_{1}=(\mathcal{P}_{1}\otimes{\rm
1l})\left(\frac{1}{d}|{\rm 1l}\rangle\\!\rangle\langle\\!\langle{\rm
1l}|\right)=\frac{1}{d}\sum_{i}|i\rangle\langle i|\otimes|x_{i}\rangle\langle
x_{i}|;\\\ \rho_{2}=(\mathcal{P}_{2}\otimes{\rm 1l})\left(\frac{1}{d}|{\rm
1l}\rangle\\!\rangle\langle\\!\langle{\rm
1l}|\right)=\frac{1}{d}\sum_{i}|i\rangle\langle i|\otimes|y_{i}\rangle\langle
y_{i}|.\end{split}$

We need to check whether
$\textrm{supp}(\rho_{1})\cap\textrm{supp}(\rho_{2})=\\{0\\}$. We have

(9)
$\begin{split}\textmd{supp}(\rho_{1})=\textrm{span}\\{|i\rangle\otimes|x_{i}\rangle\\}_{i};\\\
\textmd{supp}(\rho_{2})=\textrm{span}\\{|i\rangle\otimes|y_{i}\rangle\\}_{i}.\end{split}$

Take $|\varphi\rangle\in\textrm{supp}(\rho_{1})\cap\textrm{supp}(\rho_{2})$,
then there exist numbers $\alpha_{1},\ldots\alpha_{m}$ and
$\beta_{1},\ldots\beta_{m}$ such that

(10)
$|\varphi\rangle=\sum_{i}\alpha_{i}|i\rangle\otimes|x_{i}\rangle=\sum_{i}\beta_{i}|i\rangle\otimes|y_{i}\rangle.$

As $|x_{i}\rangle$ and $|y_{i}\rangle$ are linearly independent for all
$i=1,\ldots,k$, then $|\varphi\rangle=0$.

Therefore from Theorem 1 we obtain that $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ are perfectly distinguishable by a finite number of uses.
Non-existence of the parallel scheme follows from Corollary 1, that is
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ cannot be distinguished in the
parallel scheme as there exists a quantum state
$\rho\in\mathcal{D}(\mathcal{X})$ such that
$\rho\in\mathrm{span}\\{|x_{i}\rangle\langle y_{i}|\\}_{i}$. The algorithm
finding the state $\rho$ will be presented in Section 5.

Figure 3. Adaptive discrimination scheme

Now we focus on the adaptive scheme for perfect discrimination between
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ which is depicted in Figure 3. The top
register corresponds to the space $\mathcal{X}$ while the second and third
registers correspond to spaces $\mathcal{Y}$ and $\mathcal{Z}$ respectively
where each of these spaces is $\mathbb{C}^{3}$. Our goal is as follows. Before
the final use of the black box we would like to obtain states
$|\xi_{i}\rangle$ and $|\eta_{i}\rangle$ for which
$\left(\mathcal{P}_{1}\otimes{\rm
1l}\right)\left(|\xi_{i}\rangle\langle\xi_{i}|\right)$ and
$\left(\mathcal{P}_{2}\otimes{\rm
1l}\right)\left(|\eta_{i}\rangle\langle\eta_{i}|\right)$ are orthogonal.
Therefore we need to find processing $\Phi_{i}$ for which

(11) $\begin{split}\sum_{i}|i\rangle\langle
i|\otimes|\xi_{i}\rangle\langle\xi_{i}|&=\sum_{i}|i\rangle\langle
i|\otimes\Phi_{i}\big{(}\left(\mathcal{P}_{1}\otimes{\rm
1l}\right)(|\psi_{in}\rangle\langle\psi_{in}|)\big{)}\\\
\sum_{i}|i\rangle\langle
i|\otimes|\eta_{i}\rangle\langle\eta_{i}|&=\sum_{i}|i\rangle\langle
i|\otimes\Phi_{i}\big{(}\left(\mathcal{P}_{2}\otimes{\rm
1l}\right)(|\psi_{in}\rangle\langle\psi_{in}|)\big{)}\end{split}$

where $|\psi_{in}\rangle$ is the input state for the discrimination procedure.

The scheme for discrimination between $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$
is as follows. First, using the fact that
$\textrm{rank}\left(\mathrm{span}\\{|x_{i}\rangle\langle
y_{i}|\\}_{i}\right)<9$, we can find a nonzero matrix
$A\in\textrm{L}(\mathcal{X})$ such that

$A\perp\mathrm{span}\\{|x_{i}\rangle\langle y_{i}|\\}_{i}.$

From the singular value decomposition we can write $A=U\Sigma V^{\dagger}$.
Next, find vectors
$\tilde{|\xi\rangle},\tilde{|\eta\rangle}\in\mathcal{Y}\otimes\mathcal{Z}$ as
$\tilde{|\xi\rangle}=|U\sqrt{\Sigma}\rangle\\!\rangle$ and
$\tilde{|\eta\rangle}=|V\sqrt{\Sigma}\rangle\\!\rangle$. We can see that they
fulfill the condition

$\operatorname{Tr}_{\mathcal{Z}}\tilde{|\xi\rangle}\tilde{\langle\eta|}=A.$

Let
$|\xi\rangle=\frac{\tilde{|\xi\rangle}}{\left\|\tilde{|\xi\rangle}\right\|}$
and
$|\eta\rangle=\frac{\tilde{|\eta\rangle}}{\left\|\tilde{|\eta\rangle}\right\|}$.

Now we present the algorithm for the adaptive discrimination between
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$.

  1. (1)

As the input state to the discrimination we use the maximally entangled state
on first two registers, that is $|\psi_{in}\rangle=\frac{1}{\sqrt{d}}|{\rm
1l}_{\mathcal{X}}\rangle\\!\rangle$. Then we perform the unknown measurement
on the first register, either $\mathcal{P}_{1}$ or $\mathcal{P}_{2}$, and
obtain a label $i$. At the same time on the second register we obtain the
state either $|\psi_{i}\rangle$ or $|\varphi_{i}\rangle\in\mathcal{Y}$. Using
the property in Eq. (7) one can note that

$\left|\langle{\psi_{i}}|{\varphi_{i}}\rangle\right|=\left|\operatorname{Tr}(A)\right|^{2}=\frac{1}{2}.$

  2. (2)

Basing on the label $i$ we perform an isometric channel
$\Phi_{U_{i}}(\cdot)=U_{i}\cdot U_{i}^{\dagger}$ for
$U_{i}\in\textrm{U}(\mathcal{Y},\mathcal{Y}\otimes\mathcal{Z})$ such that

$\begin{split}U_{i}|\psi_{i}\rangle&=\mathrm{e}^{\mathrm{i}\theta_{i}}|\xi\rangle\\\
U_{i}|\varphi_{i}\rangle&=|\eta\rangle\end{split}$

where $\theta_{i}=\beta-\alpha_{i}$ for
$\langle{\psi_{i}}|{\varphi_{i}}\rangle=r_{i}\mathrm{e}^{\mathrm{i}\alpha_{i}}$
and $\langle{\xi}|{\eta}\rangle=r\mathrm{e}^{\mathrm{i}\beta}$.

  3. (3)

After performing the isometric channel we have a new third register
$\mathcal{Z}$. Then we perform the unknown measurement on the second register
$\mathcal{Y}$ and obtain a label $j$.

  4. (4)

Finally, we measure the third register by some known measurement and make a
decision whether in the black box there was either $\mathcal{P}_{1}$ or
$\mathcal{P}_{2}$.

##  5\. Numerical results

In this section we study numerically the distinguishability of random POVMs
with rank-one effects. We check the conditions for perfect distinguishability.
To choose at random a POVM of dimension $d$ with $m$ rank-one effects we take
a Haar random isometry matrix of dimension $d\times m$. Then we take
projectors onto the columns $\\{|x_{i}\rangle\\}_{i=1}^{m}$ of this matrix and
obtain a POVM with effects $\\{|x_{i}\rangle\langle x_{i}|\\}_{i=1}^{m}$.

Let $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ be POVMs with effects
$\\{|x_{i}\rangle\langle x_{i}|\\}_{i=1}^{m}$ and $\\{|y_{i}\rangle\langle
y_{i}|\\}_{i=1}^{m}$ respectively. If $m=d^{2}$, then ${\rm
1l}\in\mathrm{span}\\{|x_{i}\rangle\langle y_{i}|\\}$ and therefore any two
random rank-one POVMs cannot be distinguished perfectly by any finite number
of uses. If $d\leq m<d^{2}$, then ${\rm
1l}\not\in\mathrm{span}\\{|x_{i}\rangle\langle y_{i}|\\}$. Hence there exists
a finite number $N$ of uses which allows for perfect distinguishability iff
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are (entanglement-assisted) disjoint.
Fortunately, we have the following lemma.

######  Lemma 1.

Consider two independently chosen at random POVMs with rank-one effects. Then
with probability one they are (entanglement-assisted) disjoint.

###### Proof.

Define two states

(12) $\begin{split}\rho_{1}=(\mathcal{P}_{1}\otimes{\rm
1l})\left(\frac{1}{d}|{\rm 1l}\rangle\\!\rangle\langle\\!\langle{\rm
1l}|\right)=\frac{1}{d}\sum_{i}|i\rangle\langle i|\otimes|x_{i}\rangle\langle
x_{i}|;\\\ \rho_{2}=(\mathcal{P}_{2}\otimes{\rm 1l})\left(\frac{1}{d}|{\rm
1l}\rangle\\!\rangle\langle\\!\langle{\rm
1l}|\right)=\frac{1}{d}\sum_{i}|i\rangle\langle i|\otimes|y_{i}\rangle\langle
y_{i}|.\end{split}$

We need to check whether
$\textrm{supp}(\rho_{1})\cap\textrm{supp}(\rho_{2})=\\{0\\}$. We have

(13)
$\begin{split}\textmd{supp}(\rho_{1})=\textrm{span}\\{|i\rangle\otimes|x_{i}\rangle\\}_{i};\\\
\textmd{supp}(\rho_{2})=\textrm{span}\\{|i\rangle\otimes|y_{i}\rangle\\}_{i}.\end{split}$

Take $|\varphi\rangle\in\textrm{supp}(\rho_{1})\cap\textrm{supp}(\rho_{2})$,
then there exist $\alpha_{1},\ldots\alpha_{m}$ and $\beta_{1},\ldots\beta_{m}$
such that

(14)
$|\varphi\rangle=\sum_{i}\alpha_{i}|i\rangle\otimes|x_{i}\rangle=\sum_{i}\beta_{i}|i\rangle\otimes|y_{i}\rangle.$

As it holds that for random POVMs $|x_{i}\rangle$ and $|y_{i}\rangle$ are
linearly independent with probability one for all $i=1,\ldots,k$, then
$|\varphi\rangle=0$. ∎

arxiv Knowing that for $m<d^{2}$ two POVMs with rank-one effects can always be
distinguished perfectly by a finite number of uses, now we are interested when
it suffices to use the parallel scheme, instead of the adaptive one. From
Theorem 1 we know that if there exists a positive operator
$\rho\in\mathrm{span}(|x_{i}\rangle\langle y_{i}|)$, then we need to use the
adaptive scheme. Simple numerical calculations show that there exist pairs of
qubit POVMs with $3$ rank-one effects which can be distinguished perfectly
only by the adaptive scheme. Moreover, the probability of finding a pair of
POVMs with this property is roughly equal to $\frac{4}{10}$.

Now we proceed to studying the distinguishability of randomly chosen rank-one
POVMs in higher dimensions depending on the number of effects. We study this
problem numerically for the input dimension $d=7$ and the number of effects
$d<m<d^{2}$. The procedure is as follows. We sample $N=10^{6}$ pairs of random
POVMs in a manner explained at the beginning of this section. For further
details on sampling we refer the reader to [27, 28].

Figure 4.  Numerically estimated lower bound for the probability $p_{\rho}$ of
an event that for two randomly sampled POVMs with rank-one effects the
adaptive scenario is necessary for achieving perfect discrimination. The
probability is plotted as a function of the number of effects $m$ for the
dimension $d=7$. The number of samples per point is $N=10^{6}$.

In order to check whether the condition stated in Corollary 1 is fulfilled, we
introduce an iterative algorithm. First, we construct a projection operator
$P$ on the space $\mathrm{span}\\{|x_{i}\rangle\langle y_{i}|\\}_{i=1}^{m}$
and choose an initial operator $X\in\mathrm{L}(\mathcal{X}_{d})$. We project
$X$ onto the subspace given by $P$ obtaining some operator $Y$. We substitute
the operator $X$ with the density operator closest to $Y$ and repeat this
procedure until it converges or a predefined number of steps is reached. The
final step is to check whether the resulting operator is of full rank.

Figure 4 shows numerically found probability of finding a positive operator in
the subspace defined in Corollary 1 as a function of the number of effects of
POVMs. As we can see in the figure, as the number of effects increases, so
does this probability. This is in agreement with previous results. On the one
hand for the case when $m=d$, we have a von Neumann POVM and we have shown
[13] that such measurements can always be distinguished in a finite number of
steps in parallel. On the other hand, when we have $m=d^{2}$, then
$\mathrm{span}{\\{|x_{i}\rangle\langle y_{i}|\\}_{i}}$ gives the entire space
$\mathbb{C}^{d}$ and we are guaranteed to find the identity operator in it,
which implies that perfect distinguishability cannot be obtained for any
finite number of uses. In general, we are more likely to need to utilize the
adaptive scheme for perfect discrimination when the number of effects of POVMs
is close to $d^{2}$.

##  6\. Conclusions

In this work we studied the problem of discriminating POVMs with rank-one
effects. This class of measurements requires a more careful approach compared
to von Neumann POVMs or even general projective measurements. We show that for
this class of measurements there exist instances where the adaptive
discrimination scheme is optimal. This is contrary to von Neumann POVMs, in
which the parallel scheme is always optimal. We illustrate this result using a
special class of POVMs – the symmetric information complete POVMs. For a given
input dimension $d$ we study the discrimination between a SIC POVM and the
same SIC POVM, but with permuted effects. This allows us to construct a simple
example of two quantum channels which can be perfectly discriminated in a
finite number of steps but require an adaptive scheme. We state the
construction of such a discrimination scheme explicitly. Finally, we provide
some numerical insight on the probability of randomly sampling a pair of POVMs
with rank-one effects which can be distinguished only in the adaptive
scenario.

## Acknowledgments

AK and ZP acknowledge financial support by the Foundation for Polish Science
through TEAM-NET project (contract no. POIR.04.04.00-00-17C1/18-00). ŁP
acknowledges the support of the Polish National Science Centre under the
project number 2016/22/E/ST6/00062.

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# An effective method for computing Grothendieck point residue mappings

Shinichi Tajima  Graduate School of Science and Technology, Niigata
University,
8050, Ikarashi 2-no-cho, Nishi-ku Niigata, Japan
<EMAIL_ADDRESS>Katsusuke Nabeshima  Graduate School of
Technology, Industrial and Social Sciences, Tokushima University,
Minamijosanjima-cho 2-1, Tokushima, Japan<EMAIL_ADDRESS>
###### Abstract

Grothendieck point residue is considered in the context of computational
complex analysis. A new effective method is proposed for computing
Grothendieck point residues mappings and residues. Basic ideas of our approach
are the use of Grothendieck local duality and a transformation law for local
cohomology classes. A new tool is devised for efficiency to solve the extended
ideal membership problems in local rings. The resulting algorithms are
described with an example to illustrate them. An extension of the proposed
method to parametric cases is also discussed as an application.

###### keywords:

Grothendieck local residues mapping algebraic local cohomology transformation
law 32A27 32C36 13P10 14B15

††journal:  ††thanks: This work has been partly supported by JSPS Grant-in-Aid
for Scientific Research (C) (Nos 18K03214,18K03320).

##  1 Introduction

The theory of Grothendieck residue and duality is a cornerstone of algebraic
geometry and complex analysis (Griffiths and Harris, 1978; Grothendieck, 1957;
Hartshorne, 1966; Kunz, 2009). It has been used and applied in diverse
problems of several different fields of mathematics (Baum and Bott, 1972;
Bykov et al., 1991; Cardinal and Mourrain, 1996; Dickenstein and Sessa, 1991;
Griffiths, 1976; Lehmann, 1991; O’Brian, 1975; Perotti, 1998; Suwa, 2005). In
the global situation, methods for computing the total sum of Grothendieck
residues have been extensively studied and applied by several authors (Bykov
et al., 1991; Cattani et al., 1996; Kytmanov, 1988; Yushakov, 1984).

The concept of Grothendieck local residue together with the local duality
theory also play quite important roles in complex analysis, especially in
singularity theory (Brasselet et al., 2009; Cherveny, 2018; Corrêa et al.,
2016; Klehn, 2002; O’Brian, 1975; Suwa, 1988). Computing Grothendieck local
residues is therefore of fundamental importance. However, since the problem is
local in nature, it is difficult in general to compute Grothendieck local
residues (O’Brian, 1977). In fact, a direct use of the classical
transformation law described in (Hartshorne, 1966) only gives algorithms which
lack efficiency. Compared to the global situation, despite the importance,
much less work has been done on algorithmic aspects of computing Grothendieck
local residues (Elkadi and Mourrain, 2007; Mourrain, 1997; Ohara and Tajima,
2019a, b; Tajima and Nakamura, 2005b). Grothendieck local residues with
parameters are useful in the study of singularity theory, for example,
deformations of singularity and unfoldings of holomorphic foliations (Kulikov,
1998; Saito, 1983; Varchenko, 1986). However, to the best of our knowledge,
existing algorithm of computing Grothendieck local residues are not designed
to be able to treat parametric cases.

In this paper, we consider methods for computing Grothendieck point residues
from the point of view of complex analysis and singularity theory. We propose
a new effective method for computing Grothendieck point residues mappings and
residues, which can be extended to treat parametric cases.

Let $X\subset{\mathbb{C}}^{n}$ be an open neighborhood of the origin
$O\in{\mathbb{C}}^{n}$ and let $f_{1}(z),f_{2}(z),$ $\ldots,f_{n}(z)$ be $n$
holomorphic functions defined on $X$, where $z=(z_{1},z_{2},\ldots,z_{n})\in
X$. Assume that their common locus in $X$ is the origin $O$: $\\{z\in X\mid
f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\\}=\\{O\\}$.

Then, for a given germ $h(z)$ of holomorphic function at $O$, the Grothendieck
point residue at the origin $O$, denoted by

${\rm res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)}\right),$

of the differential form $\displaystyle{\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)}}$ can be expressed, or defined, as the integral

$\left(\frac{1}{2\pi\sqrt{-1}}\right)^{n}\int\cdots\int_{\gamma_{\epsilon}}\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)},$

where $dz=dz_{1}\wedge dz_{2}\wedge\cdots\wedge dz_{n},$ and where
$\gamma_{\epsilon}$ is a real $n$-dimensional cycle:

$\gamma_{\epsilon}=\\{z\in
X\mid|f_{1}(z)|=|f_{2}(z)|=\cdots=|f_{n}(z)|=\epsilon\\},$

with $0<\epsilon\ll 1.$ (See for instance, (Baum and Bott, 1972; Griffiths and
Harris, 1978; Tong, 1973)).

Let

$h(z)\longrightarrow{\rm
res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots f_{n}(z)}\right)$

be the Grothendieck point residue mapping that assigns to a holomorphic
function $h(z)$ the value of the Grothendieck point residue. We show that,
based on the concept of local cohomology, the use of Grothendieck local
duality and a transformation law for local cohomology classes given by J.
Lipman (Lipman, 1984) allows us to design an effective method for computing
Grothendieck local residue mappings and another one for computing Grothendieck
local residues. Note that the classical transformation law on Grothendieck
residue is of no avail for computing Grothendieck local residue mappings.
Since we compute Grothendieck local residue mappings, our method is applicable
when the holomorphic function $h(z)$ in the numerator is computable, that is
the case when the coefficients of the Taylor expansion of $h(z)$ is
computable. This is an advantage of our approach. We also show that the
proposed method can be extended to treat parametric cases. This is another
advantage of our approach.

In Section 2, we recall the transformation law for local cohomology classes
and Grothendieck local duality. In Section 3, we fix our notation and we
briefly recall our basic tool, an algorithm for computing Grothendieck local
duality. We devise, in the context of exact computation, a new tool which
plays a key role in the resulting algorithm. In Section 4, we describe the
resulting algorithm for computing Grothendieck point residue mappings and the
algorithm for computing Grothendieck point residues. In Section 5, as an
application, we generalize the proposed method to treat parametric cases and
we show, by using an example, an algorithm for computing Grothendieck point
residues associated to a $\mu$-constant deformation of quasi homogeneous
isolated hypersurface singularities.

##  2 Local analytic residues

The concept of Grothendieck point residue was introduced by A. Grothendieck in
terms of derived categories and local cohomology. In this section, we briefly
recall some basics on transformation law for local cohomology classes and
Grothendieck local duality.

Let $X\subset{\mathbb{C}}^{n}$ be an open neighborhood of the origin
$O\in{\mathbb{C}}^{n}$. Let ${\mathcal{O}}_{X}$ be the sheaf on $X$ of
holomorphic functions, and $\Omega_{X}^{n}$ the sheaf of holomorphic
$n$-forms. Let ${\mathcal{H}}_{\\{O\\}}^{n}({\mathcal{O}}_{X})$ (resp.
${\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n})$ ) denote the local cohomology
supported at $O$ of ${\mathcal{O}}_{X}$ (resp. $\Omega_{X}^{n}$).

Then, ${\mathcal{O}}_{X,O}$, the stalk at $O$ of the sheaf
${\mathcal{O}}_{X}$, and the local cohomology
${\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n})$ are mutually dual as locally
convex topological vector spaces (Bănică and Stănăşilă, 1974). The duality is
given by the point residue pairing:

${\rm res}_{\\{O\\}}(\ast,\
\ast):{\mathcal{O}}_{X,O}\times{\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n})\longrightarrow{\mathbb{C}}$

Let $F=[f_{1}(z),f_{2}(z),\ldots,f_{n}(z)]$ be an $n$-tuple of $n$ holomorphic
functions defined on $X$. Assume that their common locus $\\{z\in X\mid
f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\\}$ in $X$ is the origin $O.$ Let $I_{F}$
denote the ideal in ${\mathcal{O}}_{X,O}$ generated by
$f_{1}(z),f_{2}(z),\ldots,f_{n}(z).$ Let $\omega_{F}$ denote a local
cohomology class

$\omega_{F}=\left[\begin{array}[]{c}dz\\\ f_{1}(z)f_{2}(z)\cdots
f_{n}(z)\end{array}\right]$

in ${\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n})$, where $dz=dz_{1}\wedge
dz_{2}\wedge\cdots\wedge dz_{n},$ and $\left[\quad\right]$ stands for
Grothendieck symbol (Hartshorne, 1966; Grothendieck, 1967). Residue theory
says that, for $h(z)$ in ${\mathcal{O}}_{X,O},$ one has

${\rm res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)}\right)={\rm res}_{\\{O\\}}(h(z),\omega_{F}).$

###  2.1 Transformation law

Since $V(I_{F})\cap X=\\{O\\},$ there exists, for each $i=1,2,\ldots,n,$ a
positive integer $m_{i}$ such that $z_{i}^{m_{i}}\in I_{F}.$ There exists an
$n$-tuple of holomorphic functions $a_{i,1}(z),a_{i,2}(z),\ldots,a_{i,n}(z)$
such that

$z_{i}^{m_{i}}=a_{i,1}(z)f_{1}(z)+a_{i,2}(z)f_{2}(z)+\cdots+a_{i,n}(z)f_{n}(z),\qquad
i=1,2,\ldots,n.$

Set $A(z)=\det(a_{i,j}(z))_{1\leq i,j\leq n}.$

We have the following key lemma (Lipman, 1984).

######  Lemma 1 (Transformation law for local cohomology classes).

In ${\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n}),$ the following formula holds.

$\omega_{F}=\left[\begin{array}[]{c}A(z)dz\\\ z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots
z_{n}^{m_{n}}\end{array}\right].$

For the proof of the result above, we refer the reader to (Kunz, 2009; Lipman,
1984). Note that the formula above implies the classical transformation law

${\rm res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)}\right)={\rm
res}_{\\{O\\}}\left(\frac{h(z)A(z)dz}{z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots
z_{n}^{m_{n}}}\right)$

for point residues described in (Hartshorne, 1966). See also (Baum and Bott,
1972; Boyer and Hickel, 1997; Griffiths and Harris, 1978; Kytmanov, 1988).

###  2.2 Grothendieck local duality

We define $W_{F}$ to be the set of local cohomology classes in
${\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n})$ that are killed by $I_{F}:$

$W_{F}=\\{\omega\in{\mathcal{H}}_{\\{O\\}}^{n}(\Omega_{X}^{n})\mid
f_{1}(z)\omega=f_{2}(z)\omega=\cdots=f_{n}(z)\omega=0\\}.$

Then, according to Grothendieck local duality, the pairing

${\rm res}_{\\{O\\}}(*,\ *):{\mathcal{O}}_{X,O}/I_{F}\times
W_{F}\longrightarrow{\mathbb{C}}$

induced by the residue mapping is non-degenerate (Altman and Kleiman, 1970;
Grothendieck, 1957; Hartshorne, 1966; Lipman, 2002).

Let $\succ^{-1}$ be a local term ordering on the local ring
${\mathcal{O}}_{X,O}$ and let $\\{z^{\alpha}\mid\alpha\in\Lambda_{F}\\}$
denote the monomial basis of the quotient space ${\mathcal{O}}_{X,O}/I_{F}$
with respect to the local term ordering $\succ^{-1},$ where
$\Lambda_{F}\subset{\mathbb{N}}^{n}$ is the set of exponents $\alpha$ of basis
monomials $z^{\alpha}$.

Let $\\{\omega_{\alpha}\in W_{F}\mid\alpha\in\Lambda_{F}\\}$ denote the dual
basis of $\\{z^{\alpha}\mid\alpha\in\Lambda_{F}\\}$ with respect to the
Grothendieck point residue. Then, we have

  1. (i)

${\mathcal{O}}_{X,O}/I_{F}\cong{\rm
Span}_{{\mathbb{C}}}\\{z^{\alpha}\mid\alpha\in\Lambda_{F}\\},$

  2. (ii)

$W_{F}={\rm
Span}_{{\mathbb{C}}}\\{\omega_{\alpha}\mid\alpha\in\Lambda_{F}\\},$

  3. (iii)

${\rm
res}_{\\{O\\}}(z^{\alpha},\omega_{\beta})=\left\\{\begin{array}[]{rl}1,&\alpha=\beta,\\\
0,&\alpha\neq\beta,\qquad\alpha,\beta\in\Lambda_{F}.\\\ \end{array}\right.$

###  2.3 Residue mapping

Since $\omega_{F}$ satisfies
$f_{1}(z)\omega_{F}=f_{2}(z)\omega_{F}=\cdots=f_{n}(z)\omega_{F}=0,$ the local
cohomology class $\omega_{F}$ is in $W_{F}$. Therefore $\omega_{F}$ can be
expressed as a linear combination of the basis
$\\{\omega_{\alpha}\mid\alpha\in\Lambda_{F}\\}.$

Assume that, for the moment, we have the following expression:

$\omega_{F}=\sum_{\alpha\in\Lambda_{F}}b_{\alpha}\omega_{\alpha},\quad
b_{\alpha}\in{\mathbb{C}}.$

Now let

${\rm
NF}_{\succ^{-1}}(h)(z)=\sum_{\alpha\in\Lambda_{F}}h_{\alpha}z^{\alpha},\quad
h_{\alpha}\in{\mathbb{C}}$

be the normal form of the given holomorphic function $h(z).$ Then, we have the
following.

######  Theorem 2.

${\rm res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)}\right)=\sum_{\alpha\in\Lambda_{F}}h_{\alpha}b_{\alpha}$

######  Proof 2.1.

Since $h-{\rm NF}_{\succ^{-1}}(h)\in I_{F},$ we have

${\rm res}_{\\{O\\}}(h(z),\omega_{F})={\rm res}_{\\{O\\}}({\rm
NF}_{\succ^{-1}}(h)(z),\omega_{F}).$

Therefore,

$\displaystyle{\rm res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots
f_{n}(z)}\right)={\rm
res}_{\\{O\\}}\left(\sum_{\alpha\in\Lambda_{F}}h_{\alpha}z^{\alpha},\sum_{\beta\in\Lambda_{F}}b_{\beta}\omega_{\beta}\right),$

which is equal to

$\displaystyle\sum_{\alpha,\beta\in\Lambda_{F}}h_{\alpha}b_{\beta}{\rm
res}_{\\{O\\}}(z^{\alpha},\omega_{\beta})=\sum_{\alpha\in\Lambda_{F}}h_{\alpha}b_{\alpha}.$

This completes the proof.

##  3 Tools

Let us consider a method for computing Grothendieck point residues in the
context of symbolic computation. We start by recalling some basics on an
algorithm for computing Grothendieck local duality given in (Tajima and
Nakamura, 2009; Tajima et al., 2009).

Let $K={\mathbb{Q}}$ be the field of rational numbers and let
$z=(z_{1},z_{2},\ldots,z_{n})\in{\mathbb{C}}^{n}.$ Let $H^{n}_{[O]}(K[z])$
denote the algebraic local cohomology defined to be

$H^{n}_{[O]}(K[z])=\lim_{k\rightarrow\infty}Ext^{n}_{K[z]}(K[z]/{\mathfrak{m}}^{k},\Omega_{X}^{n}),$

where ${\mathfrak{m}}$ is the maximal ideal ${\mathfrak{m}}=\langle
z_{1},z_{2},\ldots,z_{n}\rangle$ in $K[z]=K[z_{1},z_{2},\ldots,z_{n}].$

We adopt the notation used in (Nabeshima and Tajima, 2015a, b, 2016a, 2016b)
to handle local cohomology classes. For instance, a polynomial
$\sum_{\lambda}c_{\lambda}\xi^{\lambda}$ in
$K[\xi]=K[\xi_{1},\xi_{2},\ldots,\xi_{n}]$ represents the local cohomology
class of the form
$\displaystyle\sum_{\lambda=(\ell_{1},\ell_{2},\ldots,\ell_{n})}c_{\lambda}\left[\begin{array}[]{c}1\\\
z_{1}^{\ell_{1}+1}z_{2}^{\ell_{2}+1}\cdots
z_{n}^{\ell_{n}+1}\end{array}\right].$ Note that a multiplication on
$\xi^{\beta}$ by $z^{\alpha}$ is

$z^{\alpha}\ast\xi^{\beta}=\left\\{\begin{array}[]{cl}\xi^{\beta-\alpha},&\beta\geq\alpha.\\\
0,&\text{otherwise.}\\\ \end{array}\right.$

Let $\succ$ be a term ordering on $K[\xi]$. For a local cohomology class
$\displaystyle\psi=c_{\alpha}\xi^{\alpha}+\sum_{\xi^{\alpha}\succ\xi^{\gamma}}c_{\gamma}\xi^{\gamma}$,
we call $\xi^{\alpha}$ the head monomial of $\psi$, and
$\alpha\in{\mathbb{N}}^{n}$ the head exponent of $\psi$.

Let $F=[f_{1}(z),f_{2}(z),\ldots,f_{n}(z)]$ be a list of $n$ polynomials
$f_{1},\dots,f_{n}$ in $K[z]$. We also assume as in the previous section that
there exists an open neighborhood $X$ of the origin $O$ such that their common
locus is the origin: $\\{z\in X\mid
f_{1}(z)=f_{2}(z)=\cdots=f_{n}(z)=0\\}=\\{O\\}.$

We set

$H_{F}=\\{\psi\in H^{n}_{[O]}(K[z])\mid
f_{1}(z)\ast\psi=f_{2}(z)\ast\psi=\cdots=f_{n}(z)\ast\psi=0\\}.$

###  3.1 Algorithm for computing Grothendieck local duality

In (Nabeshima and Tajima, 2017; Tajima et al., 2009), an algorithm for
computing bases of $H_{F}$ is introduced. Let $\Psi_{F}$ denote an output of
the algorithm. Then,

$W_{F}={\rm Span}_{{\mathbb{C}}}\\{\psi dz\mid\psi\in\Psi_{F}\\}$

holds. Furthermore, the algorithm computes Grothendieck local duality with
respect to the Grothendieck local residue pairing. Here we recall some basic
properties of the algorithm.

An output of the algorithm, say $\Psi_{F},$ a basis of the vector space
$H_{F}$, has the following form:

$\Psi_{F}=\left\\{{\psi_{\alpha}}\left|\,\vphantom{\psi_{\alpha}}{\psi_{\alpha}=\xi^{\alpha}+\sum_{\xi^{\alpha}\succ\xi^{\gamma}}c_{\gamma}\xi^{\gamma},\quad\alpha\in\Lambda_{F}}\right.\right\\},$

where $\Lambda_{F}\subset{\mathbb{N}}^{n}$ is the set of the head exponents of
local cohomology classes in $\Psi_{F}$.

Let $L_{F}$ denote the set of lower exponents of local cohomology classes in
$\Psi_{F}:$

$L_{F}=\left\\{{\gamma\in{\mathbb{N}}^{n}}\left|\,\vphantom{\gamma\in{\mathbb{N}}^{n}}{\exists\
\psi_{\alpha}=\xi^{\alpha}+\sum_{\xi^{\alpha}\succ\xi^{\gamma}}c_{\gamma}\xi^{\gamma}\in\Psi_{F}\text{
such that }\ c_{\gamma}\neq 0}\right.\right\\}.$

Set $E_{F}=\Lambda_{F}\cup L_{F}$ and $T_{F}=\\{\xi^{\lambda}\mid\lambda\in
E_{F}\\}.$ Now let $\ell_{F,i}={\rm max}\\{\ell\mid\xi_{i}^{\ell}\in
T_{F}\\}.$ Then Grothendieck local duality implies the following.

######  Lemma 3.

Set $m_{i}=\ell_{F,i}+1.$ Then $z_{i}^{m_{i}}\in I_{F}$ holds, where $I_{F}$
is the ideal in the local ring $K\\{z\\}$ generated by
$f_{1}(z),f_{2}(z),\ldots,f_{n}(z).$

######  Proof 3.1.

Since $z_{i}^{m_{i}}\ast\psi_{\alpha}=0$ and $\psi_{\alpha}\in\Psi_{F}$ hold,
we have $z_{i}^{m_{i}}\ast\psi=0$ for $\psi\in H_{F}.$ It follows from the
Grothendieck local duality that $z_{i}^{m_{i}}$ is in $I_{F}.$

Now let us consider the set of monomials $M_{F}$ in $K\\{z\\}$ defined to be
$M_{F}=\\{z^{\alpha}\mid\alpha\in\Lambda_{F}\\}$. Let $\succ^{-1}$ denote the
local term ordering on $K\\{z\\}$ defined as the inverse ordering of $\succ.$
Then, $M_{F}$ constitutes a monomial basis of the quotient $K\\{z\\}/I_{F}$
with respect to the local term ordering $\succ^{-1}.$ Furthermore, we have the
following result (Tajima and Nakamura, 2005a, b, 2009).

######  Theorem 4.

Let $\Psi_{F},M_{F}$ be as above. Then, $\Psi_{F}$ is the dual basis of the
basis $M_{F}$ with respect to Grothendieck local residue pairing. That is, for
$z^{\alpha}\in M_{F}$ and for $\psi_{\beta}\in\Psi_{F},$

${\rm
res}_{\\{O\\}}(z^{\alpha},\psi_{\beta}dz)=\left\\{\begin{array}[]{rl}1,&\alpha=\beta,\\\
0,&\alpha\neq\beta,\\\ \end{array}\right.$

holds.

Sketch of the proof. Since the algorithm outputs a reduced basis of $H_{F},$
we have $\Lambda_{F}\cap L_{F}=\emptyset,$ which implies the result.
$\hfill\qed$

###  3.2 A key tool

Let $m_{i}$ be an integer such that $z_{i}^{m_{i}}$ is in the ideal
$I_{F}=(f_{1},f_{2},\ldots,f_{n})$ in the local ring. Then there exist germs
$a_{i,1}(z),a_{i,2}(z),\ldots,a_{i,n}(z)$ of holomorphic functions such that

$z_{i}^{m_{i}}=a_{i,1}(z)f_{1}(z)+a_{i,2}(z)f_{2}(z)+\cdots+a_{i,n}(z)f_{n}(z),\qquad
i=1,2,\ldots,n.$

Theory of symbolic computation asserts that such $n$-tuple of holomorphic
functions can be obtained by computing syzygies in the local ring $K\\{z\\}.$
Whereas, since the cost of computation of syzygies in local rings is high, a
direct use of the classical algorithm of computing syzygy is not appropriate
in actual computations. In fact, it is difficult to obtain these holomorphic
functions. In previous papers (Nabeshima and Tajima, 2016b), the authors of
the present paper have proposed a new effective method to overcome this type
of difficulty.

We adopt the proposed method mentioned above and devise a new, much more
efficient algorithm by improving the previous algorithm presented in
(Nabeshima and Tajima, 2015b, 2016b). We start by recalling the main idea
given in (Nabeshima and Tajima, 2016b). Let $J_{F}=\langle
f_{1}(z),f_{2}(z),\ldots,$ $f_{n}(z)\rangle$ denote the ideal in the
polynomial ring $K[z]$ generated by $f_{1}(z),f_{2}(z),$ $\ldots,f_{n}(z).$
Let $J_{F,O}$ be the primary component of $J_{F}$ whose associated prime is
the maximal ideal ${\mathfrak{m}}=\left<z_{1},z_{2},\ldots,z_{n}\right>,$ and
$G_{Q}$ a Gröbner basis of the ideal quotient $Q=J_{F}:J_{F,O}\subset K[z].$
Then there is in $G_{Q}$ a polynomial, say $q(z),$ such that $q(O)\neq 0.$

Now let $r(z)\in J_{F,O}.$ Then, since $q(z)r(z)\in J_{F},$ there exists an
$n$-tuple of polynomials $p_{1}(z),p_{2}(z),\ldots,p_{n}(z)$ in $K[z]$, such
that

$q(z)r(z)=p_{1}(z)f_{1}(z)+p_{2}(z)f_{2}(z)+\cdots+p_{n}(z)f_{n}(z).$

Since, $q(O)\neq 0,$ we have a following expression in the local ring
$K\\{z\\}:$

$r(z)=\frac{p_{1}(z)}{q(z)}f_{1}(z)+\frac{p_{2}(z)}{q(z)}f_{2}(z)+\cdots+\frac{p_{n}(z)}{q(z)}f_{n}(z).$

Since $I_{F}=K\\{z\\}\otimes J_{F,O}$ and $z_{i}^{m_{i}}\in I_{F},$
$z_{i}^{m_{i}}\in J_{F,O}$ holds. Therefore, the argument above can be applied
to compute germs $a_{i,1}(z),a_{i,2}(z),\ldots,a_{i,n}(z)$ of holomorphic
functions. Note also that, since $J_{F,O}=\\{p(z)\in K[z]\mid
p(z)*\psi_{\alpha}=0,\ \psi_{\alpha}\in\Psi_{F}\\},$ the primary ideal
$J_{F,O}$ can be computed by using $\Psi_{F}.$

Let $G_{F}=\\{g_{1},g_{2},\ldots,g_{\nu}\\}$ be a Gröbner basis of $J_{F}.$
Let $R_{F}$ be a list of relations between $g_{j}$ and
$F=[f_{1},f_{2},\ldots,f_{n}]:$

$g_{j}=r_{1,j}f_{1}+r_{2,j}f_{2}+\cdots+r_{n,j}f_{n},$

where $r_{i,j}\in K[z],\ i=1,2,\ldots,n$, and $j=1,2,\ldots,\nu.$ Let $S_{F}$
be a Gröbner basis of the module of syzygies among $F:$

$s_{1}f_{1}+s_{2}f_{2}+\cdots+s_{n}f_{n}=0,$

where $s_{i}\in K[z],\ i=1,2,\ldots,n.$ Let $q$ be a polynomial in $G_{Q}$
such that $q(O)\neq 0.$

Now we are ready to present a new tool.

Algorithm 1. localexpression

Input: $G_{F},R_{F},S_{F},q,r.$
Output: $[p_{1},p_{2},\ldots,p_{n}]$ such that
$q(z)r(z)=p_{1}(z)f_{1}(z)+p_{2}(z)f_{2}(z)+\cdots+p_{n}(z)f_{n}(z).$
BEGIN

  1. step 1:

divide $qr$ by the Gröbner basis $G_{F}=\\{g_{1},g_{2},...,g_{\nu}\\}$:

$qr=e_{1}g_{1}+e_{2}g_{2}+\cdots+e_{\nu}g_{\nu};$

  2. step 2:

rewrite the relation above by using $R_{F}$:

$qr=\left(\sum_{j}r_{j,1}e_{j}\right)f_{1}+\left(\sum_{j}r_{j,2}e_{j}\right)f_{2}+\cdots+\left(\sum_{j}r_{j,2}e_{j}\right)f_{n};$

  3. step 3:

simplify the expression above by using $S_{F}$:

$q(z)r(z)=p_{1}(z)f_{1}(z)+p_{2}(z)f_{2}(z)+\cdots+p_{n}(z)f_{n}(z);$

return $[p_{1},p_{2},\ldots,p_{n}];$
END

######  Example 5 ($E_{12}$ singularity).

Let $f(x,y)=x^{3}+y^{7}+xy^{5}$ and let $F=[\frac{\partial f}{\partial
x}(x,y),$ $\frac{\partial f}{\partial y}(x,y)].$ Note that $f(x,y)$ is a semi
quasi-homogeneous function with respect to the weight vector $(7,3).$ Let
$\succ$ be the weighted degree lexicographical ordering on $K[\xi,\eta]$ with
respect to the weight vector $(7,3),$ where $\xi,\eta$ correspond to $x,y.$

Then, $\dim_{K}(H_{F})=12,$ the Milnor number at the origin $(0,0)$ of the
curve $\\{(x,y)\in{\mathbb{C}}^{2}\mid f(x,y)=0\\}.$ The algorithm for
computing Grothendieck local duality, mentioned in this section, outputs a
basis $\Psi_{F}$ that consists of the following 12 local cohomology classes;

$1,\eta,\xi,\eta^{2},\xi\eta,\eta^{3},\xi\eta^{2},\eta^{4},\eta^{5}-\frac{1}{3}\xi^{2},\xi\eta^{3},\xi\eta^{4}-\frac{5}{7}\eta^{6}+\frac{5}{21}\xi^{2}\eta,$

$\xi\eta^{5}-\frac{5}{7}\eta^{7}-\frac{1}{3}\xi^{3}+\frac{5}{21}\xi^{2}\eta^{2}.$

Note for instance that the local cohomology class
$\left[\begin{array}[]{c}1\\\
xy^{6}\end{array}\right]-\frac{1}{3}\left[\begin{array}[]{c}1\\\
x^{3}y\end{array}\right]$ represented by
$\psi_{(0,5)}=\eta^{5}-\frac{1}{3}\xi^{2}$ above acts on a holomorphic
function $h(x,y)=\sum_{(i,j)}c_{(i,j)}x^{i}y^{j}$ by

${\rm res}_{O}(h(x,y),\psi_{(0,5)}dx\wedge
dy)=c_{(0,5)}-\frac{1}{3}c_{(2,0)}.$

The output implies that

$\Lambda_{F}=\\{(0,0),(0,1),(0,2),(1,0),(0.3),(1,1),(0,4),(1,2),(0,5),(1,3),(1,4),$

$(1,5)\\}$

and $M_{F}=\\{x^{i}y^{j}\mid(i,j)\in\Lambda_{F}\\}$ is the monomial basis of
the quotient space $K\\{x,y\\}/I_{F}$ with respect to the local term ordering
$\succ^{-1}$ on $K\\{x,y\\},$ where $I_{F}$ denote the ideal in $K\\{x,y\\}$
generated by $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}$.
Furthermore $W_{F}=\\{\psi dx\wedge dy\mid\psi\in\Psi_{F}\\}$ is the dual
basis of the monomial basis $M_{F}$ with respect to the Grothendieck local
residue pairing. Since $\lambda_{F}=(3,7)$, we have $x^{4},y^{8}\in I_{F}.$

Let $J_{F}$ be the ideal in $K[x,y]$ generated by the two polynomials
$\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}.$ Let $J_{F,O}$
be the primary component of $J_{F}$ whose associated prime is the maximal
ideal $\left<x,y\right>.$ A Gröbner basis of the ideal quotient
$J_{F,O}:J_{F}$ is

$3125x+151263,\ 25y+147.$

Set $q(x,y)=25y+147.$ Then, the algorithm localexpression outputs the
following:

$\displaystyle q(x,y)x^{4}$ $\displaystyle=$
$\displaystyle(49x^{2}+25/3x^{2}y-49/3y^{5})\frac{\partial f}{\partial
x}+(-5/3xy^{2}+7/3y^{4})\frac{\partial f}{\partial y},$ $\displaystyle
q(x,y)y^{8}$ $\displaystyle=$ $\displaystyle 25y^{4}\frac{\partial f}{\partial
x}+(-15x+21y^{2})\frac{\partial f}{\partial y}.$

##  4 Algorithms

Let $\tau_{F}$ denote the local cohomology class in $H_{F}$ defined to be

$\tau_{F}=\left[\begin{array}[]{c}1\\\ f_{1}(z)f_{2}(z)\cdots
f_{n}(z)\end{array}\right].$

Since $\omega_{F}=\tau_{F}dz,$ the local cohomology class $\tau_{F}$ is the
kernel function of the point residue mapping.

Let

$q(z)z_{i}^{m_{i}}=p_{i,1}(z)f_{1}(z)+p_{i,2}(z)f_{2}(z)+\cdots+p_{i,n}(z)f_{n}(z),\
i=1,2,\ldots,n,$

and set ${\rm Det}(z)={\rm det}(p_{i,j}(z))_{1\leq i,j\leq n}.$

Let $I_{M}$ be the ideal in $K[z]$ generated by
$z_{1}^{m_{1}},z_{2}^{m_{2}},\ldots,z_{n}^{m_{n}}.$ Let $u(z)\in K[z]$ be a
polynomial such that $u(z)q(z)-1\in I_{M}$.

Since $A(z)=\det(p_{i,j}(z)/q(z))_{1\leq i,j\leq n}$ is equal to
$\frac{1}{q(z)^{n}}{\rm Det}(z),$ the transformation law implies the following

$\displaystyle\tau_{F}=\left[\begin{array}[]{c}u(z)^{n}{\rm Det}(z)\\\
z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots z_{n}^{m_{n}}\end{array}\right].$

Let $\lambda_{F}=(\ell_{F,1},\ell_{F,2},\ldots,\ell_{F,n}).$ Since
$m_{i}=\ell_{F,i}+1,$ the formula above can be rewritten as
$\tau_{F}=u(z)^{n}{\rm Det}(z)\ast\xi^{\lambda_{F}}.$

Note that, according to an algorithm in (Sato and Suzuki, 2009) discovered by
Y. Sato and A. Suzuki, the inverse $u(z)$ of $q(z)$ in $K[z]/I_{M}$ can be
obtained by using Gröbner basis computation.

The following algorithm computes a representation of the local cohomology
class $\tau_{F}$, the kernel function of the point residue mapping.

Algorithm 2. tau

Input:
$V=[z_{1},z_{2},\ldots,z_{n}],\succ,F=[f_{1}(z),f_{2}(z),\ldots,f_{n}(z)].$
/* $V$: a list of variables, $\succ$: a term order */
Output: $\tau_{F}=\sum_{\alpha\in\Lambda_{F}}b_{\alpha}\psi_{\alpha}.$
BEGIN

  1. step 1:

compute a basis $\Psi_{F}=\\{\psi_{\alpha}\mid\alpha\in\Lambda_{F}\\}$ of the
space $H_{F}$;
/* $\Lambda_{F}$: the set of head terms of $\Psi_{F}$ */

  2. step 2:

compute $\ell_{F,i}=\max\\{\ell|\xi_{i}^{\ell}\in T_{F}\\}$ and set
$m_{i}=\ell_{F,i}+1,\ i=1,2,\ldots,n$;
/* $T_{F}=\\{\xi^{\lambda}\mid\lambda\in E_{F}\\}$ */

  3. step 3:

compute a Gröbner basis of the ideal

$J_{F,O}=\\{p(z)\in K[z]\mid p(z)\ast\psi_{\alpha}=0,\alpha\in\Lambda_{F}\\};$

  4. step 4:

compute $G_{F},R_{F},S_{F}$;
/* notations are from subsection 3.2*/

  5. step 5:

compute a Gröbner basis $G_{Q}$ of the quotient ideal $Q=J_{F}:J_{F,O}$ and
choose a polynomial $q(z)$ from $G_{Q}$ such that $q(O)\neq 0$;

  6. step 6:

compute

$q(z)z_{i}^{m_{i}}=p_{i,1}(z)f_{1}(z)+p_{i,2}(z)f_{2}(z)+\cdots+p_{i,n}(z)f_{n}(z),\
(i=1,2,\ldots,n),$

by using the algorithm localexpression;

  7. step 7:

compute ${\rm Det}(z)={\rm det}(p_{i,j}(z))_{1\leq i,j\leq n}$ and set ${\rm
ND}={\rm NF}_{I_{M}}({\rm Det}(z)),$ the normal form of ${\rm Det}(z)$ with
respect to $I_{M}$;

  8. step 8:

compute a Gröbner basis of the ideal in $K[z,u]$ generated by

$1-q(z)u,z_{1}^{m_{1}},z_{2}^{m_{2}},\ldots,z_{n}^{m_{n}}$

with respect to an elimination ordering to eliminate $u$;

  9. step 9:

choose a polynomial of degree one with respect to $u$, of the form
$cu+poly(z)$, from the Gröbner basis of step 8 and set

${\rm Den}=(-c)^{n},\ {\rm NU}={\rm NF}_{I_{M}}(poly(z)^{n}),\ {\rm Num}={\rm
NF}_{I_{M}}({\rm ND}\times{\rm NU});$

  10. step10:

compute $\psi={\rm Num}\ast\xi^{\lambda_{F}}$ and set ${\rm
Coeff}=\\{c_{\alpha}\mid\alpha\in\Lambda_{F}\\}$;
/* $c_{\alpha}$ is the coefficient of a term $\xi^{\alpha}$ of $\psi$,
$\alpha\in\Lambda_{F}.$ */

return $[\Lambda_{F},\Psi_{F},{\rm Coeff},{\rm Den}];$
END

The return of the algorithm above means

$\tau_{F}=\frac{1}{{\rm
Den}}\sum_{\alpha\in\Lambda_{F}}c_{\alpha}\psi_{\alpha}.$

Note that, since,

${\operatorname{res}}_{O}(h(z)\tau_{F}dz)=\frac{1}{{\rm
Den}}\sum_{\alpha\in\Lambda_{F}}b_{\alpha}{|res}_{O}(h(z)\psi_{\alpha}dz)$

holds, the output of the algorithm above completely describes the Grothendieck
point residue mapping

$h(z)\longrightarrow{\rm
res}_{\\{O\\}}\left(\frac{h(z)dz}{f_{1}(z)f_{2}(z)\cdots f_{n}(z)}\right).$

Let ${\rm Res}_{F}={\bf tau}(V,\succ,F)$ be the output of the algorithm tau.
The following algorithm residues evaluates the value of Grothendieck point
residue.

Algorithm 3. residues

Input: $h\in K[z],$ ${\rm Res}_{F}.$
Output: ${\rm res}_{\\{O\\}}(h(z)\tau_{F}dz).$
BEGIN

  1. step 1:

compute the normal form of $h$ by using $\Psi_{F}$, i.e., $\displaystyle{\rm
NF}_{\succ}(h)(z)=\sum_{\alpha\in\Lambda_{F}}h_{\alpha}z^{\alpha}$;

  2. step 2:

compute $\displaystyle{\rm
sum}=\sum_{\alpha\in\Lambda_{F}}h_{\alpha}c_{\alpha};$

return $\displaystyle\frac{{\rm sum}}{{\rm Den}}$ ;
END

Note that ${\rm NF}_{\succ}(h)$ is computed by the algorithms given in (Tajima
and Nakamura, 2009; Tajima et al., 2009). The algorithm is free from standard
bases computation. All the algorithms given in the present paper are
implemented in a computer algebra system Risa/Asir(Noro and Takeshima, 1992)).

######  Example 6 ($E_{12}$ singularity).

Let us continue the computation. Since step 1 to step 6 are done, we start
from step 7. From

$\left(\begin{array}[]{cc}p_{1,1}&p_{1,2}\\\
p_{2,1}&p_{2,2}\end{array}\right)=\left(\begin{array}[]{cc}25/3x^{2}y+49x^{2}&-5/3xy^{2}+7/3y^{4}\\\
25y^{4}&-15x+21y^{2}\end{array}\right),$

we have the determinant

${\rm
Det}=(-125y-735)x^{3}+(175y^{3}+1029y^{2})x^{2}+(125/3y^{6}+245y^{5})x-175/3y^{8}-343y^{7}.$

A Gröbner basis of the ideal in $K[x,y,u]$ generated by
$1-uq(x,y),x^{4},y^{8}$ with respect to a elimination ordering $u\succ x,y$ is
$\\{x^{4},y^{8},-6103515625y^{7}+35888671875y^{6}-211025390625y^{5}+1240829296875y^{4}$
$-7296076265625y^{3}+42900928441875y^{2}-252257459238225y-21804125746715$
$2161u+1483273860320763\\}.$

We have

${\rm Num}=(6654091109227055694580078125y^{7}-391260557222550874841308593$
$75y^{6}+230061207646859914406689453125y^{5}-1352759900963536296711333984$
$375y^{4}+7954228217665593424662643828125y^{3}-46770861919873689337016345$
$709375y^{2}+275012668088857293301656112771125y-1617074488362480884613$
$737943094215)x^{3}+(-322085690705603880169365234375y^{7}+189386386134895$
$0815395867578125y^{6}-11135919504731830794527701359375y^{5}+6547920668$
$7823165071822883993125y^{4}-385017735324400210622318557879575y^{3}+22639$
$04283707473238459233120331901y^{2})x^{2}+(1559028730662456311233878190312$
$5y^{7}-91670889362952431100552037590375y^{6}+539024829454160294871245981$
$031405y^{5})x-754634761235824412819744373443967y^{7}$
and ${\rm Den}=(218041257467152161)^{2}.$

Since $\displaystyle b_{\alpha}=\frac{c_{\alpha}}{{\rm Den}},$ we have
$\tau_{F}=\displaystyle{\frac{1}{{\rm Den}}({\rm Num}\ast(\xi^{3}\eta^{7}))}.$

Therefore,

$\tau_{F}=30517578125/218041257467152161-1220703125/1483273860320763\eta+4$
$8828125/10090298369529\eta^{2}-1953125/68641485507\eta^{2}+78125/466948881\eta^{4}-3$
$125/3176523\eta^{5}+125/21609\eta^{6}-5/147\eta^{7}-9765625/1441471195647\xi+390625/9$
$805926501\xi\eta-15625/66706983\xi\eta^{2}+625/453789\xi\eta^{3}-25/3087\xi\eta^{4}+1/21\xi\eta^{5}+3125/9529569\xi^{2}-125/64827\xi^{2}\eta+5/441\xi^{2}\eta^{2}-1/63\xi^{3}.$

This yields

$\tau_{F}=\sum_{0\leq i,j\leq 5}b_{i,j}\psi_{i,j},$

where $b_{0,0}=30517578125/218041257467152161,$
$b_{0,1}=-1220703125/1483273860320763,\\\
b_{0,2}=48828125/10090298369529,b_{0,3}=-1953125/68641485507,\\\
b_{0,4}=78125/466948881,b_{0,5}=-3125/3176523,\\\
b_{1,0}=-9765625/1441471195647,b_{1,1}=390625/9805926501,\\\
b_{1,2}=-15625/66706983,b_{1,3}=625/453789,b_{1,4}=-25/3087,b_{1,5}=1/21.$
and

$\psi_{0,0}=1,\ \psi_{0,1}=\eta,\ \psi_{0,2}=\eta^{2},\ \psi_{0,3}=\eta^{3},\
\psi_{0,4}=\eta^{4},\ \psi_{0,5}=\eta^{5}-\dfrac{1}{3}\xi^{2},\\\
\psi_{1,0}=\xi,\ \psi_{1,1}=\xi\eta,\psi_{1,2}=\xi\eta^{2},\
\psi_{1,3}=\xi\eta^{3},\\\
\psi_{1,4}=\xi\eta^{4}-\dfrac{5}{7}\eta^{6}+\dfrac{5}{21}\xi^{2}\eta,\
\psi_{1,5}=\xi\eta^{5}-\dfrac{5}{7}\eta^{7}-\dfrac{1}{3}\xi^{3}+\dfrac{5}{21}\xi^{2}\eta^{2}.$

Let ${\rm NF}_{\succ}(h)(x,y)=\sum_{(i,j)\in\Lambda_{F}}h_{i,j}x^{i}y^{j}.$
Then,

${\rm res}_{\\{O\\}}(h(x,y),\tau_{F}dx\wedge
dy)=\sum_{(i,j)\in\Lambda_{F}}h_{i,j}b_{i,j}.$

We have for instance, $\displaystyle{{\rm res}_{\\{O\\}}\left(\frac{dx\wedge
dy}{\frac{\partial f}{\partial x}\frac{\partial f}{\partial
y}}\right)=\frac{30517578125}{218041257467152161}.}$

Recall that , as local cohomology class $\omega_{F}=\tau_{F}dx\wedge dy$ is in
${\mathcal{H}}_{\\{(0,0)\\}}^{2}(\Omega_{X}^{2}),$ the cohomology class
$\tau_{F}$ defines the residue mapping

${\rm
res}_{\\{O\\}}(\ast,\tau_{F}):{\mathcal{O}}_{X,O}\longrightarrow{\mathbb{C}}.$

Therefore, the formula above is valid for germs of holomorphic functions
$h(x,y)$. More precisely, for a germ of holomorphic function $\displaystyle
h(x,y)=\sum_{(i,j)}c_{i,j}x^{i}y^{j}$, we have

${\rm res}_{\\{O\\}}(h(x,y),\tau_{F}dx\wedge
dy)=c_{0,0}b_{0,0}+c_{0,1}b_{0,1}+c_{0,2}b_{0,2}+c_{1,0}b_{1,0}+c_{0,3}b_{0,3}+c_{1,1}b_{1,1}+c_{0,4}b_{0,4}+c_{1,2}b_{1,2}+(c_{0,5}-\frac{1}{3}c_{2,0})b_{0,5}+c_{1,3}b_{1,3}+(c_{1,4}-\frac{5}{7}c_{0,6}+\frac{5}{21}c_{2,1})b_{1,4}+(c_{1,5}-\frac{1}{3}c_{3,0}-\frac{5}{7}c_{0,7}+\frac{5}{21}c_{2,2})b_{1,5}.$

##  5  $\mu$-constant deformation

In this section, we consider a $\mu$-constant deformation of a quasi
homogeneous singularity, a family of semi-quasi homogeneous isolated
hypersurface singularities (Greuel, 1986; Lê and Ramanujam, 1976). We give, as
an application of the algorithms presented in the previous section, an
algorithm for computing Grothendieck point residues associated to a
$\mu$-constant deformation of a quasi homogeneous isolated hypersurface
singularity. The keys of the resulting algorithm are the use of parametric
local cohomology systems and parametric Gröbner systems (comprehensive Gröbner
systems).

Let $w=(w_{1},w_{2},\ldots,w_{n})\in{\mathbb{N}}^{n}$ be a weight vector for
$z=(z_{1},z_{2},\ldots,z_{n}).$ Let $d_{w}(z^{\lambda})$ denote the weighted
degree of a monomial $z^{\lambda}=z_{1}^{\ell_{1}}z_{2}^{\ell_{2}}\cdots
z_{n}^{\ell_{n}}$ defined to be

$d_{w}(z^{\lambda})=\ell_{1}w_{1}+\ell_{2}w_{2}+\cdots+\ell_{n}w_{n}.$

######  Definition 7.

  1. (1)

A non-zero polynomial $f_{0}$ is called a weighted homogeneous (or quasi
homogeneous) polynomial of type $(d,w)$, if all monomials of $f_{0}$ have the
same weighted degree $d$ with respect to the weight vector $w$, that is
$f_{0}=\sum_{d_{w}(z^{\lambda})=d}c_{\lambda}z^{\lambda}$ where
$c_{\lambda}\in K$.

  2. (2)

A polynomial $f(z)=f_{0}(z)+g(z)$ is called a semi weighted homogeneous (or
semi quasi homogeneous) polynomial of type $(d,w)$, if

  3. (i)

$f_{0}$ is weighted homogeneous of type $(d,w)$, and $f_{0}(z)=0$ has an
isolated singularity at the origin $O$, and

  4. (ii)

$g(z)=\sum_{d_{w}(z^{\beta_{j}})>d}b_{j}z^{\beta_{j}},$ where $b_{j}$ are
coefficients.

Let $t=(t_{1},t_{2},\ldots,t_{m})$ denote a set of new indeterminates, and let
$T=\\{t\mid t\in{\mathbb{C}}^{m}\\}.$ Let

$f_{t}(z)=f_{0}(z)+g(z,t),\ \mbox{with}\ \
g(z,t)=\sum_{d_{w}(z^{\beta_{j}})>d}t_{j}z^{\beta_{j}}$

be a family of semi weighted homogeneous polynomials in $K(t)[z]$, where $t\in
T$ is regarded as a deformation parameter. Then $f_{t}$ is a $\mu$-constant
deformation of $f_{0}.$

Set $F=[\frac{\partial f}{\partial z_{1}},\frac{\partial f}{\partial
z_{2}},\ldots,\frac{\partial f}{\partial z_{n}}].$ Let $I_{F}$ denote a family
of ideals in $K(t)\\{z\\}$ generated by $F$ with the parameter $t\in T$ and
let

$H_{F}=\left\\{{\psi\in H_{\\{O\\}}^{n}(K(t)[z])}\left|\,\vphantom{\psi\in
H_{\\{O\\}}^{n}(K(t)[z])}{\frac{\partial f}{\partial
z_{1}}\ast\psi=\frac{\partial f}{\partial z_{2}}\ast\psi=\cdots=\frac{\partial
f}{\partial z_{n}}\ast\psi=0}\right.\right\\}.$

Let $\succ$ be a term ordering on
$K(t)[\xi]=K(t)[\xi_{1},\xi_{2},\ldots,\xi_{n}]$ compatible with the weight
vector $w.$ It is known, for semi weighted homogeneous cases, that the set of
leading exponents $\Lambda_{F}$ is independent of $t$ and thus so is the
corresponding basis monomial set $M_{F}.$ In our previous papers (Nabeshima
and Tajima, 2015c, b), an algorithm for computing a basis $\Psi_{F}$ of
$H_{F}$ is given. The algorithm also computes Grothendieck local duality as in
the non parametric cases. The other steps, from step 3 to step 10 in the
algorithm tau  are also executable by using parametric Gröbner systems. The
step 1 and step 2 of the algorithm residues are also executable.

Here we give an example of computation.

######  Example 8 ($E_{12}$ singularity).

Let us consider $f=x^{3}+y^{7}+txy^{5}$ ( $t\neq 0$).
step 1: A basis $\Psi_{F}$ of the vector space $H_{F}$ with respect to a term
ordering $\succ$ compatible with the weight $w=(7,3)$ is

$\left\\{1,\eta,\eta^{2},\xi,\eta^{3},\xi\eta,\eta^{4},\xi\eta^{2},\eta^{5}-\frac{t}{3}\xi^{2},\xi\eta^{3},\xi\eta^{4}-\frac{5t}{7}\eta^{6}+\frac{5t^{2}}{21}\xi^{2}\eta,\right.$

$\left.\xi\eta^{5}-\frac{t}{3}\xi^{3}-\frac{5t}{7}\eta^{7}+\frac{5t^{2}}{21}\xi^{2}\eta^{2}\right\\}.$

The set $\Lambda_{F}$ is

$\Lambda_{F}=\\{(0,0),(0,1),(0,2),(1,0),(0.3),(1,1),(0,4),(1,2),(0,5),(1,3),(1,4),$

$(1,5)\\}.$

step 2: $x^{4},y^{8}\in I_{F}.$

step 5: $q(x,y)=147+25t^{3}y\in J_{F}:J_{F,O}.$

step 6: $\left(\begin{array}[]{c}q(x,y)x^{4}\\\ q(x,y)y^{8}\end{array}\right)$
$=\left(\begin{array}[]{cc}(25/3t^{3}y+49)x^{2}-49/3ty^{5},&-5/3t^{3}y^{2}x+7/3t^{2}y^{4}\\\
25t^{2}y^{4}&-15tx+21y^{2}\end{array}\right)\left(\begin{array}[]{c}\frac{\partial
f}{\partial x}\\\ \frac{\partial f}{\partial y}\end{array}\right).$

step 7: ${\rm Det}(x,y)$ is

$(-125t^{4}y-735t)x^{3}+(175t^{3}y^{3}+1029y^{2})x^{2}+(125/3t^{5}y^{6}+245t^{2}y^{5})x-175/3t^{4}y^{8}-343ty^{7}.$

step 8: A Gröbner basis of $\langle x^{4},y^{8},1-q(x,y)u\rangle$ is

$\\{y^{8},x^{4},-6103515625t^{21}y^{7}+35888671875t^{18}y^{6}-211025390625t^{15}y^{5}+12408292$
$96875t^{12}y^{4}-7296076265625t^{9}y^{3}+42900928441875t^{6}y^{2}-252257459238225t^{3}y-218041257467152161u+1483273860320763\\}.$

step 9: We have

${\rm Den}=(218041257467152161)^{2},$

$poly(x,y)=-6103515625t^{21}y^{7}+35888671875t^{18}y^{6}-211025390625t^{15}y^{5}+1$
$240829296875t^{12}y^{4}-7296076265625t^{9}y^{3}+42900928441875t^{6}y^{2}-252257459238$
$225t^{3}y+1483273860320763,$

${\rm NU}=-72425481460974755859375000t^{21}y^{7}+372629102116715118896484375$
$t^{18}y^{6}-1878050674668244199238281250t^{15}y^{5}+920244830587439657626757812$
$5t^{12}y^{4}-43288316830833161494762687500t^{9}y^{3}+19090147722397424219190345$
$1875t^{6}y^{2}-748333790717979029392261531350t^{3}y+220010134471085834641324$
$8902169,$

and

${\rm
Num}=(6654091109227055694580078125t^{22}y^{7}-39126055722255087484130859$
$375t^{19}y^{6}+230061207646859914406689453125t^{16}y^{5}-13527599009635362967113$
$33984375t^{13}y^{4}+7954228217665593424662643828125t^{10}y^{3}-46770861919873689$
$337016345709375t^{7}y^{2}+27501266808885729330165611277112tt^{4}y-16170744883$
$62480884613737943094215t)x^{3}+(-322085690705603880169365234375t^{15}y^{7}+1$
$893863861348950815395867578125t^{12}y^{6}-11135919504731830794527701359375$
$t^{9}y^{5}+65479206687823165071822883993125t^{6}y^{4}-38501773532440021062231855$
$7879575t^{3}y^{3}+2263904283707473238459233120331901y^{2})x^{2}+(155902873066245$
$63112338781903125t^{8}y^{7}-91670889362952431100552037590375t^{5}y^{6}+539024829$
$454160294871245981031405t^{2}y^{5})x-754634761235824412819744373443967ty^{7}.$

As an output we thus have

$\displaystyle\tau_{F}=\sum_{0\leq i,j\leq 5}b_{i,j}\psi_{i,j},$

where
$b_{0,0}=30517578125t^{22}/218041257467152161,\\\
b_{0,1}=-1220703125t^{19}/1483273860320763,\\\
b_{0,2}=48828125t^{16}/10090298369529,b_{0,3}=-1953125t^{13}/68641485507,\\\
b_{0,4}=78125t^{10}/466948881,b_{0,5}=-3125t^{7}/3176523,\\\
b_{1,0}=-9765625t^{15}/1441471195647,\\\
b_{1,1}=390625t^{12}/9805926501,b_{1,2}=-15625t^{9}/66706983,\\\
b_{1,3}=625t^{6}/453789,b_{1,4}=-25t^{3}/3087,b_{1,5}=1/21.$
and

$\psi_{0,0}=1,\ \psi_{0,1}=\eta,\ \psi_{0,2}=\eta^{2},\ \psi_{0,3}=\eta^{3},\
\psi_{0,4}=\eta^{4},\ \psi_{0,5}=\eta^{5}-\dfrac{t}{3}\xi^{2},\\\
\psi_{1,0}=\xi,\ \psi_{1,1}=\xi\eta,\ \psi_{1,2}=\xi\eta^{2},\
\psi_{1,3}=\xi\eta^{3},\
\psi_{1,4}=\xi\eta^{4}-\dfrac{5t}{7}\eta^{6}+\dfrac{5t^{2}}{21}\xi^{2}\eta,\\\
\psi_{1,5}=\xi\eta^{5}-\dfrac{5t}{7}\eta^{7}-\dfrac{t}{3}\xi^{3}+\dfrac{5t^{2}}{21}\xi^{2}\eta^{2}.$

We have, for instance,

$\displaystyle{\rm res}_{\\{O\\}}\left(\frac{dx\wedge dy}{\frac{\partial
f}{\partial x}\frac{\partial f}{\partial
y}}\right)=\frac{30517578125}{218041257467152161}t^{22}.$

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