p}_{X}$}\\!/\\!\raisebox{-1.37775pt}{$T_{\mathscr{R}}(\Phi_{Y}f)+T_{\mathscr{L}}(\Phi_{Y}f)$}\right.}={\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
p}_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\Phi^{\prime}_{Y}T_{\mathscr{R}}f+T_{\mathscr{L}}f$}\right.}\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{\Phi_{Y}^{\prime}}T^{1}_{\mathscr{A}}f.$

  2. ii.

Suppose $T^{1}_{\mathscr{A}}f\sim T^{1}_{\mathscr{A}}{\tilde{f}}$. Using the
corresponding morphism of algebras we get $T^{1}_{\mathscr{A}}(\Phi_{Y}\circ
f\circ\Phi_{X})=T^{1}_{\mathscr{A}}{\tilde{f}}$, i.e.
$T_{\mathscr{A}}(\Phi_{Y}\circ
f\circ\Phi_{X})=T_{\mathscr{A}}{\tilde{f}}\subset R^{\oplus p}_{X}$.

######  Theorem 7.4.

Let
$R_{X}={\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[x]]$}\\!/\\!\raisebox{-1.37775pt}{$J$}\right.}$,
with $\Bbbk=\bar{\Bbbk}$ and $char(\Bbbk)=0$. Suppose
$\sqrt{I}={\mathfrak{m}}$.

  1. 1.

The $\mathscr{A}$-type of $f$ is determined by the mixed module type of
$T^{1}_{\mathscr{A}^{(0)}}f$.

  2. 2.

If $f$ is $\mathscr{A}$-finite and not stable then the $\mathscr{A}$-type
of $f$ is determined by the mixed module type of $T^{1}_{\mathscr{A}}f$.

More precisely, if $T^{1}_{\mathscr{A}^{(0)}}f_{0}\sim
T^{1}_{\mathscr{A}^{(0)}}f_{1}$, then
$f_{0}\stackrel{{\scriptstyle\mathscr{A}^{(0)}}}{{\sim}}f_{1}$. (And similarly
in case 2.)
Proof. By the example above we can assume
$T_{\mathscr{A}^{(0)}}f_{0}=T_{\mathscr{A}^{(0)}}f_{1}\subset R^{\oplus
p}_{X}$ (resp. $T_{\mathscr{A}}f_{0}=T_{\mathscr{A}}f_{1}\subset R^{\oplus
p}_{X}$). The proof is similar to the ${\mathscr{K}}\\!$-case and is
inductive. We apply a sequence of transformations
$g_{d}\in\mathscr{A}^{(0)}(jet_{d}(R_{X}))$ and verify
$jet_{d}f\stackrel{{\scriptstyle g_{d}\cdots
g_{1}}}{{\sim}}jet_{d}{\tilde{f}}$. Moreover, the transformation can be chosen
to satisfy $jet_{d-1}(g_{d})=Id$.

Fix any $d\geq 1$ and replace $R_{X}$ by the Artinian ring
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(x)^{d+1}$}\right.}$.
Assume $jet_{d-1}(f_{0})=jet_{d-1}(f_{1})$.

  1. Step 1.

As in the ${\mathscr{K}}\\!$-case we define $f_{t}:=f_{0}+t(f_{1}-f_{0})$ and
$R_{X,t}:=R_{X}\otimes\Bbbk[t]$,
$T_{\mathscr{A}_{t}}:=T_{\mathscr{A}}\otimes\Bbbk[t]$. We claim:
$\partial_{t}f_{t}\in T_{\mathscr{A}^{(j_{d})}_{t}}f_{t}$ for
$t\in\Bbbk^{1}\setminus$(finite set), i.e. this holds over the fraction ring
$\Bbbk[t][S^{-1}]$. Here $j_{d}\geq-1$ and $\lim_{d\to\infty}j_{d}=\infty$.

We have:
$\partial_{t}f_{t}=f_{1}-f_{0}\in(T_{\mathscr{A}^{(0)}}f_{0}+T_{\mathscr{A}^{(0)}}f_{1})\cap{\mathfrak{m}}^{d}\cdot
R^{\oplus p}_{X}=T_{\mathscr{A}^{(0)}}f_{0}\cap{\mathfrak{m}}^{d}\cdot
R^{\oplus p}_{X}$. (For part two one gets: $\partial_{t}f_{t}\in
T_{\mathscr{A}}f_{0}\cap{\mathfrak{m}}^{d}\cdot R^{\oplus p}_{X}$.)

By the “mixed Artin-Rees” lemma, [Kerner.21, §7], one has
$T_{\mathscr{A}}f_{0}\cap{\mathfrak{m}}^{d}\cdot R^{\oplus p}_{X}\subset
T_{\mathscr{A}^{(j_{d})}}f_{0}$, where $\lim_{d}j_{d}=\infty$.

Therefore, as in the ${\mathscr{K}}\\!$-case, it is enough to verify
$T_{\mathscr{A}^{(j_{d})}_{t}}f_{t}=T_{\mathscr{A}^{(j_{d})}_{t}}f_{0}$ for
$t\in\Bbbk^{1}\setminus$(finite set).

     * •

We have $T_{\mathscr{R}^{(j_{d})}_{t}}f_{t}\subseteq(1-t)\cdot
T_{\mathscr{R}^{(j_{d})}_{t}}f_{0}+t\cdot
T_{\mathscr{R}^{(j_{d})}_{t}}f_{1}\subseteq
T_{\mathscr{A}^{(j_{d})}_{t}}f_{0}$. In addition

(32) $T_{\mathscr{L}_{t}}f_{t}\subseteq\\{\sum_{i,j\geq 0}q_{i}(f_{0})\cdot
v_{j}(f_{1})\\}\subseteq{T_{\mathscr{L}_{t}}f_{0}+T_{\mathscr{L}_{t}}f_{1}}\subseteq{T_{\mathscr{A}_{t}}f_{0}}.$

(Note that $R_{X}$ is Artinian now.) And similarly
$T_{\mathscr{L}^{(j_{d})}_{t}}f_{t}\subseteq
T_{\mathscr{A}^{(j_{d})}_{t}}f_{0}$.

     * •

As in the ${\mathscr{K}}\\!$-case we have:
$T_{\mathscr{A}^{(j_{d})}_{t}}f_{0}\subseteq
T_{\mathscr{A}^{(j_{d})}_{t}}f_{t}+t\cdot T_{\mathscr{A}^{(j_{d})}_{t}}f_{0}$.
These modules are finitely generated over $R_{Y,t}$, because $R_{X}$is
Artinian. Localize at $(y,t)$ to get:
$(T_{\mathscr{A}^{(j_{d})}_{t}}f_{0})_{(t,y)}\subseteq(T_{\mathscr{A}^{(j_{d})}_{t}}f_{t})_{(t,y)}$.

Similarly one gets
$(T_{\mathscr{A}^{(j_{d})}_{t}}f_{1})_{(t,y)}\subseteq(T_{\mathscr{A}^{(j_{d})}_{t}}f_{t})_{(t,y)}$.

     * •

As in the ${\mathscr{K}}\\!$-case take the quotient
$M:={\footnotesize\left.\raisebox{1.37775pt}{$T_{\mathscr{A}^{(j_{d})}_{t}}f_{0}$}\\!/\\!\raisebox{-1.37775pt}{$T_{\mathscr{A}^{(j_{d})}_{t}}f_{t}$}\right.}$.
This is an $R_{Y,t}$-module. Thus: $M_{(t,y)}=0,$ and $M_{(1-t,y)}=0$. Hence
$Supp(M)\subset V(y)\subset Spec(R_{Y,t})$ is a finite subset.

  2. Step 2.

    1. i.

(The case $\boldsymbol{\Bbbk=\mathbb{C}}$.) As in the ${\mathscr{K}}\\!$-case
take a path inside $\mathbb{C}_{t}\cong
V(y)\subset(\mathbb{C}^{p}_{y},o)\times(\mathbb{C}^{1}_{t},o)$ avoiding the
finite subset $Supp(M)$. Along this path one has $\partial_{t}f_{t}\in
T_{\mathscr{A}^{(j_{d})}_{t}}f_{t}$. Now use part $\mathscr{A}$ of theorem 4.3
to get $f_{0}\stackrel{{\scriptstyle\mathscr{A}^{(j_{d})}}}{{\sim}}f_{1}$.
Note that the ring $R_{X}$ is Artinian, therefore the formal and ordinary
equivalence coincide.

For part 2 we remark that the $\mathscr{A}$-trivialization preserves the
origins of $(\Bbbk^{n},o)$, $(\Bbbk^{p},o)$, as $f_{0},f_{1}$ are
$\mathscr{A}$-finite, i.e. their instability locus is just one point.

    2. ii.

(The general case, $\boldsymbol{\Bbbk=\bar{\Bbbk},char(\Bbbk)=0}$.) By the
same Lefschetz-type arguments (and lemma 2.2) we get
$f_{0}\stackrel{{\scriptstyle\mathscr{A}^{(j_{d})}}}{{\sim}}f_{1}$.

Now combine these constructed transformations to get the (formal) group
element $g:=\lim_{d}(g_{d}\cdots g_{1})\in\mathscr{A}^{(0)}$ (resp.
$g:=\lim_{d}(g_{d}\cdots g_{1})\in\mathscr{A}$), satisfying $g{\tilde{f}}=f$.
$\blacksquare$

###  7.3. The case of arbitrary characteristic

Let $R_{X}$ be one of
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[x]]$}\\!/\\!\raisebox{-1.37775pt}{$J$}\right.}$,
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\\{x\\}$}\\!/\\!\raisebox{-1.37775pt}{$J$}\right.}$,
${\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk\langle
x\rangle$}\\!/\\!\raisebox{-1.37775pt}{$J$}\right.}$, here $\Bbbk$ is any
field. If $char(\Bbbk)>0$ then we assume the $jet_{0}$ condition of §2.1. Take
a map $f:(\Bbbk^{n},o)\to(\Bbbk^{p},o)$, thus $0\neq f\in{\mathfrak{m}}\cdot
R^{\oplus p}_{X}$. The order of $f$ is the largest $ord(f)\in\mathbb{N}$
satisfying $(f)\subseteq{\mathfrak{m}}^{ord(f)}$. If $ord(f)\leq 2$ then we
take ${\mathfrak{m}}^{ord(f)-2}=R_{X}$.

######  Theorem 7.5.

  1. 1.

(${\mathscr{K}}\\!$-case) Suppose an ideal
$\mathfrak{a}\subseteq{\mathfrak{m}}^{2}$ satisfies:
$\mathfrak{a}^{2}\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}\cdot
T_{\mathscr{R}}f+{\mathfrak{m}}\cdot(f)\cdot R^{\oplus p}_{X}$. Then the
${\mathscr{K}}\\!$-type of $f$ is determined by the $\Bbbk$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(f)+\mathfrak{a}\cdot\mathfrak{a}_{\mathscr{R}}$}\right.}$.

  2. 2.

($\mathscr{A}$-case, $\Bbbk$-infinite) Suppose an ideal
$\mathfrak{a}\subseteq{\mathfrak{m}}^{2}$ satisfies:
$\mathfrak{a}^{2}\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq\mathfrak{a}\cdot T_{\mathscr{R}}f+f^{\\#}(y)^{2}\cdot
T_{\mathscr{L}}f$. Then the $\mathscr{A}$-type of $f$ is determined by the
$\Bbbk$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$f^{\\#}(y)+\mathfrak{a}^{2}$}\right.}$.

  3. 3.

($\mathscr{R}$-case) Suppose an ideal $\mathfrak{a}\subseteq{\mathfrak{m}}$
satisfies: $\mathfrak{a}^{2}\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}\cdot T_{\mathscr{R}}f$. Then
the $\mathscr{R}$-type of $f$ is determined by the $\Bbbk[f]$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}\cdot\mathfrak{a}_{\mathscr{R}}$}\right.}$.

Here is the explicit form of the statement. Fix some maps
$f,{\tilde{f}}\in{\mathfrak{m}}\cdot R^{\oplus p}_{X}$ and ideals
$\mathfrak{a}^{f},\mathfrak{a}^{\tilde{f}}\subseteq{\mathfrak{m}}^{2}$.

  * •

($\boldsymbol{{\mathscr{K}}\\!}$) Suppose
$(\mathfrak{a}^{f})^{2}\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{f}\cdot
T_{\mathscr{R}}f+{\mathfrak{m}}\cdot(f)\cdot R^{\oplus p}_{X}$ and
$(\mathfrak{a}^{\tilde{f}})^{2}\cdot{\mathfrak{m}}^{ord({\tilde{f}})-2}\cdot
R^{\oplus p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{\tilde{f}}\cdot
T_{\mathscr{R}}{\tilde{f}}+{\mathfrak{m}}\cdot({\tilde{f}})\cdot R^{\oplus
p}_{X}$. If the $\Bbbk$-algebras are isomorphic,
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(f)+\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}$}\right.}\cong{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$({\tilde{f}})+\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}^{\tilde{f}}_{\mathscr{R}}$}\right.}$,
then $f\stackrel{{\scriptstyle{\mathscr{K}}\\!}}{{\sim}}{\tilde{f}}$.

  * •

($\boldsymbol{\mathscr{A}}$) Suppose
$(\mathfrak{a}^{f})^{2}\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{f}\cdot
T_{\mathscr{R}}f+f^{\\#}(y)^{2}\cdot T_{\mathscr{L}}f$ and
$(\mathfrak{a}^{\tilde{f}})^{2}\cdot{\mathfrak{m}}^{ord({\tilde{f}})-2}\cdot
R^{\oplus p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{\tilde{f}}\cdot
T_{\mathscr{R}}{\tilde{f}}+{\tilde{f}}^{\\#}(y)^{2}\cdot
T_{\mathscr{L}}{\tilde{f}}$. If the $\Bbbk$-algebras are isomorphic,
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$f^{\\#}(y)+(\mathfrak{a}^{f})^{2}$}\right.}\cong{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{${\tilde{f}}^{\\#}(y)+(\mathfrak{a}^{\tilde{f}})^{2}$}\right.}$,
then $f\stackrel{{\scriptstyle\mathscr{A}}}{{\sim}}{\tilde{f}}$.

  * •

($\boldsymbol{\mathscr{R}}$) Suppose
$(\mathfrak{a}^{f})^{2}\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{f}\cdot T_{\mathscr{R}}f$ and
$(\mathfrak{a}^{\tilde{f}})^{2}\cdot{\mathfrak{m}}^{ord({\tilde{f}})-2}\cdot
R^{\oplus p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{\tilde{f}}\cdot
T_{\mathscr{R}}{\tilde{f}}$. Suppose the $\Bbbk[f]$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}$}\right.}$
is isomorphic to the $\Bbbk[{\tilde{f}}]$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}^{\tilde{f}}_{\mathscr{R}}$}\right.}$.
Then $f\stackrel{{\scriptstyle\mathscr{R}}}{{\sim}}{\tilde{f}}$.

We do not assume isolated singularities or that the germ $V(f)\subset X$ is a
complete intersection.
Proof.

  1. 1.

Given an isomorphism of $\Bbbk$-algebras
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(f)+\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}$}\right.}\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$({\tilde{f}})+\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}^{\tilde{f}}_{\mathscr{R}}$}\right.}$
take its representative $\phi:R_{X}\stackrel{{\scriptstyle}}{{\to}}R_{X}$.
Thus $\phi$ is invertible (hence an isomorphism of $\Bbbk$-algebras) and sends
$(f)+\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}$ to
$({\tilde{f}})+\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}_{\mathscr{R}}({\tilde{f}})$.
Therefore after a coordinate change ($\mathscr{R}$-equivalence) we can assume:

(33)
$(f)+\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}=({\tilde{f}})+\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}_{\mathscr{R}}({\tilde{f}})\subset
R_{X}.$

Note that the assumption
$\mathfrak{a}^{2}(f)\cdot{\mathfrak{m}}^{ord(f)-2}\cdot R^{\oplus
p}_{X}\subseteq{\mathfrak{m}}\cdot\mathfrak{a}^{f}\cdot
T_{\mathscr{R}}f+{\mathfrak{m}}(f)\cdot R^{\oplus p}_{X}$ is preserved under
${\mathscr{K}}\\!$-equivalence, see [Kerner.21, §5].

By the initial assumption
$\mathfrak{a}^{f},\mathfrak{a}^{\tilde{f}}\subseteq{\mathfrak{m}}^{2}$. Then
equation (33) gives: $rank(f)=rank({\tilde{f}})$. By
${\mathscr{K}}\\!$-equivalence we can assume
$f=(x_{1},\dots,x_{r},{\mathfrak{f}})$, with
$({\mathfrak{f}})\subseteq(x_{r+1},\dots,x_{n})$. (And similarly for
${\tilde{f}}$.) Therefore the whole question is reduced to the case
$(f),({\tilde{f}})\subseteq{\mathfrak{m}}^{2}$.

Now apply $GL(p,R_{X})$ transformations to $f$ and ${\tilde{f}}$ to get:
$f-{\tilde{f}}\in({\mathfrak{m}}\cdot(f)+\mathfrak{a}^{f}\cdot\mathfrak{a}^{f}_{\mathscr{R}})\cdot
R^{\oplus p}_{X}$. Finally, by [Kerner.21, §5], we have
${\mathscr{K}}\\!f\supseteq\\{f\\}+({\mathfrak{m}}\cdot(f)+\mathfrak{a}^{f}\cdot\mathfrak{a}^{f}_{\mathscr{R}})\cdot
R^{\oplus p}_{X}$. In particular, ${\tilde{f}}\in{\mathscr{K}}\\!f$.

  2. 2.

As in the ${\mathscr{K}}\\!$-case we lift the isomorphism
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$f^{\\#}(y)+\mathfrak{a}^{2}(f)$}\right.}\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{${\tilde{f}}^{\\#}(y)+\mathfrak{a}^{2}({\tilde{f}})$}\right.}$
to an isomorphism of $\Bbbk$-algebras
$\phi:R_{X}\stackrel{{\scriptstyle}}{{\xrightarrow[\,\smash{\raisebox{4.95134pt}{$\scriptstyle\sim$}}\,]{}}}R_{X}$
sending $f^{\\#}(y)+(\mathfrak{a}^{f})^{2}$ to
${\tilde{f}}^{\\#}(y)+(\mathfrak{a}^{\tilde{f}})^{2}$. Then by a coordinate
change ($\mathscr{R}$) we can assume
$f^{\\#}(y)+(\mathfrak{a}^{f})^{2}={\tilde{f}}^{\\#}(y)+(\mathfrak{a}^{\tilde{f}})^{2}\subset
R_{X}$. Therefore ${\tilde{f}}^{\\#}(y)\in f^{\\#}(y)+(\mathfrak{a}^{f})^{2}$.
Applying $GL(p,\Bbbk)$ transformations to ${\tilde{f}}$ we can assume:
$f-{\tilde{f}}\in f^{\\#}(y)^{2}T_{\mathscr{L}}f+(\mathfrak{a}^{f})^{2}\cdot
R^{\oplus p}_{X}$. Finally, by [Kerner.21, §7], we have
$\mathscr{A}{\tilde{f}}\supseteq\\{f\\}+f^{\\#}(y)^{2}\cdot
T_{\mathscr{L}}f+(\mathfrak{a}^{f})^{2}\cdot R^{\oplus p}_{X}$. In particular,
$f\in\mathscr{A}g$.

  3. 3.

We have the isomorphism
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}$}\right.}\cong{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}^{\tilde{f}}_{\mathscr{R}}$}\right.}$,
compatible with the isomorphism $\Bbbk[f]\cong\Bbbk[{\tilde{f}}]$. The later
isomorphism (after an automorphism of $\Bbbk[{\tilde{f}}]$) can be taken as
$f_{i}\to{\tilde{f}}_{i}$. Then
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}$}\right.}\cong{\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}_{\mathscr{R}}({\tilde{f}})$}\right.}$
becomes an isomorphism of $\Bbbk[f]$-algebras.

Lift it to an isomorphism $R_{X}\stackrel{{\scriptstyle\phi}}{{\to}}R_{X}$.
Therefore by a coordinate change ($\mathscr{R}$-equivalence) we can assume:
$\mathfrak{a}^{f}\cdot\mathfrak{a}_{\mathscr{R}}^{f}=\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}_{\mathscr{R}}({\tilde{f}})\subset
R_{X}$. Moreover, we have: ${\tilde{f}}_{i}={\tilde{f}}_{i}\cdot 1\in
f_{i}+\mathfrak{a}^{f}\cdot\mathfrak{a}^{f}_{\mathscr{R}}\cdot R^{\oplus
p}_{X}$. By [Kerner.21, §4] we get
$\mathscr{R}f\supseteq\\{{\tilde{f}}\\}+\mathfrak{a}^{\tilde{f}}\cdot\mathfrak{a}^{\tilde{f}}_{\mathscr{R}}\cdot
R^{\oplus p}_{X}$. In particular, ${\tilde{f}}\in\mathscr{R}f$. $\blacksquare$

######  Example 7.6.

Take $p=1$ and assume $J=0$, i.e. $X\cong(\Bbbk^{n},o)$.

  1. i.

(The ${\mathscr{K}}\\!$-case.) Suppose
$Jac(f)\cdot{\mathfrak{m}}\cdot\mathfrak{a}+{\mathfrak{m}}\cdot(f)\supseteq\mathfrak{a}^{2}\cdot{\mathfrak{m}}^{ord(f)-2}$.
Then the ${\mathscr{K}}\\!$-type of $f$ is determined by the $\Bbbk$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(f)+\mathfrak{a}\cdot
Jac(f)$}\right.}$. The singularity can be non-isolated, we do not assume
$\sqrt{\mathfrak{a}}={\mathfrak{m}}$.

As a particular case take $\mathfrak{a}={\mathfrak{m}}^{d}$ with $d\geq 2$.
Suppose
$Jac(f)\cdot{\mathfrak{m}}^{d+1}+{\mathfrak{m}}\cdot(f)\supseteq{\mathfrak{m}}^{2d+ord(f)-2}$.
Then the ${\mathscr{K}}\\!$-type of $f$ is determined by
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(f)+{\mathfrak{m}}^{d}\cdot
Jac(f)$}\right.}$. Compare this to [Gr.Ph.17, Theorem 2.2] (for
$R_{X}=\Bbbk[[x]]$):

If
$Jac(f)\cdot{\mathfrak{m}}^{2}+{\mathfrak{m}}\cdot(f)\supseteq{\mathfrak{m}}^{\frac{d}{2}+ord(f)}$
then ${\mathscr{K}}\\!f$ is determined by the $\Bbbk$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$(f)+{\mathfrak{m}}^{d}\cdot
Jac(f)$}\right.}$.
Their assumption implies (is stronger than) the condition
$Jac(f)\cdot{\mathfrak{m}}^{\frac{3d}{2}}+{\mathfrak{m}}^{\frac{3d}{2}-1}\cdot(f)\supseteq{\mathfrak{m}}^{2d+ord(f)}$,
which is much stronger than ours.

  2. ii.

(The $\mathscr{R}$-case.) Suppose
$Jac(f)\cdot{\mathfrak{m}}\cdot\mathfrak{a}\supseteq\mathfrak{a}^{2}\cdot{\mathfrak{m}}^{ord(f)-2}$.
Then the $\mathscr{R}$-type of $f$ is determined by the $\Bbbk[f]$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{a}\cdot
Jac(f)$}\right.}$. Again, we do not assume
$\sqrt{\mathfrak{a}}={\mathfrak{m}}$.

As a particular case take take $\mathfrak{a}={\mathfrak{m}}^{d}$ with $d\geq
2$. Suppose
$Jac(f)\cdot{\mathfrak{m}}^{d+1}\supseteq{\mathfrak{m}}^{2d+ord(f)-2}$. Then
the $\mathscr{R}$-type of $f$ is determined by
${\footnotesize\left.\raisebox{1.37775pt}{$R_{X}$}\\!/\\!\raisebox{-1.37775pt}{${\mathfrak{m}}^{d}\cdot
Jac(f)$}\right.}$. This strengthens (and extends) [Gr.Ph.17, Theorem 2.4] (for
$R_{X}=\Bbbk[[x]]$).

The $\mathscr{R},{\mathscr{K}}\\!$-cases for $p>1$ and the $\mathscr{A}$-case
are new.

######  Remark 7.7.

  1. i.

One would like a stronger statement, e.g. (for $p=1$) of the form “the
${\mathscr{K}}\\!$-type is determined by the $\Bbbk$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R$}\\!/\\!\raisebox{-1.37775pt}{$(f)+{\mathfrak{m}}^{d}\cdot
Jac(f)$}\right.}$, where $d$ depends on $\Bbbk$ and $dim(R_{X})$, but not on
$f$”. This is impossible due to the following example:

(34) $char(\Bbbk)=p,R_{X}=\Bbbk[[x,y]]\text{ and
}f(x,y)=x^{p+1}+y^{pN+1},{\tilde{f}}(x,y)=f(x,y)+x^{p}y^{pd},\quad\text{ for
}\quad N>pd.$

Here $Jac(f)=Jac({\tilde{f}})$ and ${\mathfrak{m}}^{j}\cdot
Jac({\tilde{f}})\supset{\mathfrak{m}}^{d}\cdot x^{p}\ni x^{p}y^{pd}$ for any
$j\leq d$. Therefore $(f)+{\mathfrak{m}}^{j}\cdot
Jac({\tilde{f}})=({\tilde{f}})+{\mathfrak{m}}^{j}\cdot Jac(g)$ for $j\leq d$.
But $f\not\stackrel{{\scriptstyle{\mathscr{K}}\\!}}{{\sim}}g$, e.g. because
the monomial $x^{p}y^{pd}$ lies under the Newton diagram of $f$.

  2. ii.

(For $p=1$) Recall that for $R=\mathbb{C}\\{x\\}$ the $\mathscr{R}$-type of
$f$ is not determined just by the $\Bbbk$-algebra structure of
${\footnotesize\left.\raisebox{1.37775pt}{$R$}\\!/\\!\raisebox{-1.37775pt}{$Jac(f)$}\right.}$.
Moreover, the $\mathscr{R}$-type is not determined by the $\Bbbk$-algebra
${\footnotesize\left.\raisebox{1.37775pt}{$R$}\\!/\\!\raisebox{-1.37775pt}{$Jac(f)^{d}$}\right.}$,
for any $d$. Indeed, suppose
$f\stackrel{{\scriptstyle\mathscr{R}}}{{\not\sim}}{\tilde{f}}$, but
$f\stackrel{{\scriptstyle{\mathscr{K}}\\!}}{{\sim}}{\tilde{f}}$ via
${\tilde{f}}(x)=c\cdot f(\phi(x))$, with $c\in\Bbbk$. (An explicit example is
$f_{t}(x_{1},x_{2})=x^{4}_{2}+x^{5}_{1}+t\cdot x^{2}_{2}x^{3}_{1}$, see
[Gr.Lo.Sh, pg.133].) Then $Jac(f_{t})^{d}=Jac(f_{0})^{d}$ for all $d\geq 1$.

##  Appendix A Separability of unfoldings

Below we assume $\Bbbk=\bar{\Bbbk}$. To an unfolding $F=(f_{t}(x,u),t)$ we
associate the element $[f_{t}]\in T^{1}_{\mathscr{G}_{t}}f_{o}$. Define the
group action

(35) $G:=Aut_{(\Bbbk^{r},o)}:=Aut_{\Bbbk}(\mathbbm{K})\circlearrowright
R^{\oplus p}_{X,t}\quad\text{ by }\quad q(x,t)\to q(x,\phi(t)).$

This action preserves the embedding $T_{\mathscr{G}_{t}}f_{o}\subseteq
R^{\oplus p}_{X,t}$ and hence descends to the action $G\circlearrowright
T^{1}_{\mathscr{G}_{t}}f_{o}$. We get the group orbit $[Gf_{t}]\subseteq
T^{1}_{\mathscr{G}_{t}}f_{o}$ and the orbit map $G\stackrel{{\scriptstyle
f_{t}}}{{\to}}T^{1}_{\mathscr{G}_{t}}f_{o}$ by $g\to[g\cdot f_{t}]$.

Take an ideal $\mathfrak{b}\subset R_{X,t}$ satisfying
$\sqrt{\mathfrak{b}}=(x,t)$ and pass to the finite jets,
$jet_{\mathfrak{b}}(R_{X,t}):={\footnotesize\left.\raisebox{1.37775pt}{$R_{X,t}$}\\!/\\!\raisebox{-1.37775pt}{$\mathfrak{b}$}\right.}$.
This is a finite-dimensional $\Bbbk$-vector space. Similarly one has
$jet_{\mathfrak{b}}(R^{\oplus p}_{X,t})$,
$jet_{\mathfrak{b}}(T^{1}_{\mathscr{G}_{t}}f_{o}):={\footnotesize\left.\raisebox{1.37775pt}{$R^{\oplus
p}_{X,t}$}\\!/\\!\raisebox{-1.37775pt}{$T_{\mathscr{G}}f_{o}+\mathfrak{b}\cdot
R^{\oplus p}_{X,t}$}\right.}$ and
$jet_{\mathfrak{b}}(G):=Aut_{\Bbbk}({\footnotesize\left.\raisebox{1.37775pt}{$\Bbbk[[t]]$}\\!/\\!\raisebox{-1.37775pt}{$\Bbbk[[t]]\cap\mathfrak{b}$}\right.})$.
We get the (regular) action of the affine algebraic group on the affine space,
$jet_{\mathfrak{b}}(G)\circlearrowright
jet_{\mathfrak{b}}(T^{1}_{\mathscr{G}}f_{o})$.

######  Definition A.1.

(Suppose $\Bbbk=\bar{\Bbbk}$.) The orbit map $G\stackrel{{\scriptstyle
f_{t}}}{{\to}}T^{1}_{\mathscr{G}_{t}}f_{o}$ is called separable if all its
finite jets, $\\{jet_{\mathfrak{b}}(G)\stackrel{{\scriptstyle
jet_{\mathfrak{b}}(f_{t})}}{{\to}}jet_{\mathfrak{b}}T^{1}_{\mathscr{G}}f_{o}\\}_{\mathfrak{b}}$,
are separable as morphisms onto their images.

(The morphism $jet_{\mathfrak{b}}(f_{t})$ is called separable if the
corresponding field extensions are separable, see e.g. [F.-S. R.].)

Take the tangent space, $T_{G}:=(t)\cdot Der_{(\Bbbk^{r},o)}$, and its finite
jets, $\\{jet_{\mathfrak{b}}T_{G}\\}_{\mathfrak{b}}$. Take the map from the
jet(image tangent space) to the tangent space of the jet-orbit,
$jet_{\mathfrak{b}}(T_{G}f_{t})\to Tjet_{\mathfrak{b}}(Gf_{t})$. We write
$T_{G}f_{t}\twoheadrightarrow[T(Gf_{t})]\subset T^{1}_{\mathscr{G}}f_{o}$ if
the surjectivity holds for all the finite jets. Recall the general fact: the
map $f_{t}:G\to T^{1}_{\mathscr{G}}f_{o}$ is separable (as a morphism onto its
image) iff $T_{G}f_{t}\twoheadrightarrow[TGf_{t}]$. (Namely, the surjectivity
holds for all the finite jets.)

######  Definition A.2.

The unfolding $F$ is called separable if the map $G\stackrel{{\scriptstyle
f_{t}}}{{\to}}T^{1}_{\mathscr{G}}f_{o}$ is separable.

If a field $\Bbbk$ is not algebraically closed then we call $F$ separable if
$\bar{\Bbbk}\otimes F$ is separable.

######  Example A.3.

  1. i.

For $char(\Bbbk)=0$ the group orbit map is always separable. Hence any
unfolding is separable.
ii. The (in)separability is preserved under the $\mathscr{G}_{t}$-equivalence.
For example, for $\mathscr{G}=\mathscr{R}$ and any $\mathfrak{b}$ we have the
commutative diagram of algebraic varieties
$\begin{matrix}jet_{\mathfrak{b}}(G)\stackrel{{\scriptstyle
f_{t}}}{{\to}}jet_{\mathfrak{b}}(T^{1}_{\mathscr{R}}f_{o})\\\ \downarrow
f_{t}\circ\phi\ \ \|\\\
jet_{\mathfrak{b}}T^{1}_{\mathscr{R}}(f_{o}\circ\phi)=jet_{\mathfrak{b}}T^{1}_{\mathscr{R}}(f_{o})\end{matrix}$.

  2. iii.

A trivial unfolding is separable. Indeed, we can take $F=(f_{o},t)$, then
$G\stackrel{{\scriptstyle f_{o}}}{{\to}}T^{1}_{\mathscr{G}}f_{o}$ is the zero
map. Hence $[T_{G}f_{t}]=0$. And thus trivially
$T_{G}f_{t}\twoheadrightarrow[T(Gf_{t})]$.

######  Lemma A.4.

An unfolding $F$ is inseparable iff
$f\stackrel{{\scriptstyle\mathscr{G}_{t}}}{{\sim}}f_{o}+t^{d}f_{d}+\cdots$,
where $char(\Bbbk)\mid d$ and $f_{d}\not\in T_{\mathscr{G}}f_{o}$.

Proof. $\Lleftarrow$ We can assume $f=f_{o}+t^{d}f_{d}+\cdots$ with
$char(\Bbbk)\mid d$ and $f_{d}\not\in T_{\mathscr{G}}f_{o}$. Then
$T_{G}f_{t}\subseteq(t)^{d+1}\cdot R^{\oplus p}_{X,t}$. Thus for
$\mathfrak{b}=(t^{d+1},(x)^{N})$ we get $jet_{\mathfrak{b}}(T_{G}f_{t})=0$.
But $Gf_{t}\ni f_{o}+c^{d}t^{d}f_{d}+\cdots$ for any $c\in\Bbbk$. Therefore
the subvariety $[jet_{\mathfrak{b}}(Gf_{t})]\subset
jet_{\mathfrak{b}}T^{1}_{\mathscr{G}}f_{o}$ is of positive dimension. Thus
$T(jet_{\mathfrak{b}}Gf_{t})\neq 0$. Therefore the map
$T_{G}f_{t}\to[TG(f_{t})]\subset T^{1}_{\mathscr{G}}f_{o}$ cannot be
surjective.

$\Rrightarrow$ Suppose $f\sim f_{o}+t^{d}f_{d}+\cdots$ with $char(\Bbbk)\nmid
d$ and $f_{d}\not\in T_{\mathscr{G}}f_{o}$. Then
$jet_{\mathfrak{b}}T_{G}f_{t}=Span_{\Bbbk[[t]]}\\{t^{d-1}f_{d}+\cdots\\}$ and
$Tjet_{\mathfrak{b}}(Gf_{t})=Span_{\Bbbk[[t]]}\\{t^{d-1}f_{d}+\cdots\\}$.
Hence $F$ is separable.

Otherwise for each $d$ we get: $f\sim f_{o}+t^{d}f_{d}+\cdots$ with $f_{d}\in
T_{\mathscr{G}}f_{o}$. Then $F$ is formally trivial. Trivialize it to get:
$(f_{o},t)$ is separable. $\blacksquare$

## References

  * [A.G.V.] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps. Volume 1. Classification of critical points, caustics and wave fronts. Reprint of the 1985 edition. Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2012
  * [A.G.L.V.] V. I. Arnold, V.V. Goryunov, O.V. Lyashko, V.A. Vasil’ev, Singularity theory. I. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. VI, Encyclopaedia Math. Sci., 6, Springer, Berlin, 1993; MR1230637]. Springer-Verlag, Berlin, 1998. iv+245 pp
  * [Artin] M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. IHES, 36, (1969), 23-58. DOI: 10.1007/BF02684596
  * [B.K.16] G. Belitski, D. Kerner, Group actions on filtered modules and finite determinacy. Finding large submodules in the orbit by linearization, C. R. Math. Acad. Sci. Soc. R. Can. 38 (2016), no. 4, 113–153.
  * [B.G.K.22] A.-F. Boix, G.-M. Greuel, D. Kerner, Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.), Math. Z. 301 (2022), no. 3, 2415–2463.
  * [Boubakri] Y. Boubakri, Hypersurface singularities in positive characteristic, PhD thesis, 2009.
  * [Damon84] J. Damon, The unfolding and determinacy theorems for subgroups of $\mathscr{A}$ and ${\mathscr{K}}\\!$. Mem. Amer. Math. Soc. 50 (1984), no. 306, x+88 pp.
  * [Ebeling] W. Ebeling, Functions of several complex variables and their singularities. Translated from the 2001 German original by Philip G. Spain. Graduate Studies in Mathematics, 83. American Mathematical Society, Providence, RI, 2007. xviii+312 pp.
  * [Eisenbud] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995.
  * [F.-S. R.] W. R. Ferrer Santos, A. Rittatore, Actions and invariants of algebraic groups. Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2017. xx+459 pp.
  * [Fischer] G. Fischer, Complex analytic geometry. Lecture Notes in Mathematics, Vol. 538. Springer-Verlag, Berlin-New York, 1976. vii+201 pp.
  * [Gabrielov] A. M. Gabrièlov, Formal relations among analytic functions. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1056–1090.
  * [Gaffney-Hauser.85] T. Gaffney, H. Hauser, Characterizing singularities of varieties and of mappings. Invent. Math. 81 (1985), no. 3, 427–447.
  * [Gaffney-du Plessis.82] T. Gaffney, A. du Plessis, More on the determinacy of smooth map-germs. Invent. Math. 66 (1982), no. 1, 137–163.
  * [Greuel.18] G. M. Greuel, Singularities in positive characteristic: equisingularity, classification, determinacy. Singularities, algebraic geometry, commutative algebra, and related topics, 37–53, Springer, Cham, 2018
  * [Gre.Krö.90] G.-M. Greuel, H. Kröning, _Simple singularities in positive characteristic_ , Math. Z. 203 (1990), 339–354.
  * [Gr.Lo.Sh] G.-M. Greuel, C. Lossen, E. Shustin, Introduction to singularities and deformations. Springer Monographs in Mathematics. Springer, Berlin, 2007. xii+471 pp.
  * [Gr.Ng.16] G.-M. Greuel, H. D. Nguyen, Right simple singularities in positive characteristic. J. Reine Angew. Math. 712 (2016), 81–106.
  * [Gr.Ph.17] G.-M. Greuel, T.H. Pham, Mather-Yau Theorem in Positive Characteristic, J. Algebraic Geom. 26 (2017), no. 2, 347–355.
  * [Gr.Ph.19] G.-M. Greuel, T.H. Pham Finite determinacy of matrices and ideals in arbitrary characteristics, J. Algebra 530 (2019), 195–214.
  * [Jacobson] N. Jacobson, Lie algebras. Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ix+331 pp.
  * [Kerner.21] D. Kerner, Detecting modules inside the group orbits $\mathscr{R},{\mathscr{K}}\\!,\mathscr{A}$. The case of arbitrary characteristic., arXiv:2111.02715
  * [Martinet.1976] J. Martinet, Déploiements versels des applications différentiables et classification des applications stables. In Singularités d’applications différentiables (Sém., Plans-sur-Bex, 1975), volume 535 of Lecture Notes in Math., pages 1–44. Springer, Berlin, 1976.
  * [Martinet] J. Martinet, Singularities of smooth functions and maps, Lecture Note Series 58, Cambridge University Press, 1982, 256 pp.
  * [Mather] J.N. Mather, Stability of $C^{\infty}$ mappings. I. The division theorem. Ann. of Math. (2) 87, 1968, 89–104.
Stability of $C^{\infty}$ mappings. II. Infinitesimal stability implies
stability. Ann. of Math. (2) 89, 1969, 254–291.
Stability of $C^{\infty}$ mappings. III. Finitely determined map-germs. Publ.
Inst. Hautes Études Sci. Publ. Math. No. 35, 1968, 279–308.
Stability of $C^{\infty}$ mappings. IV. Classification of stable germs by
R-algebras. Publ. Inst. Hautes Études Sci. Publ. Math. No. 37, 1969, 223–248.

  * [Mather-Yau] J. N. Mather, S. S. T. Yau, Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69 (1982), no. 2, 243–251.
  * [Mond-Montaldi.91] D. Mond, J. Montaldi, Deformations of maps on complete intersections, Damon’s ${\mathscr{K}}\\!_{V}$-equivalence and bifurcations. Brasselet, Jean-Paul (ed.), Singularities. Papers of the international congress ‘Singularities in geometry and topology’, held in Lille (France), 3-8 June, 1991. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 201, 263–284 (1994).
  * [Mo. N.B.] D. Mond, J.-J. Nuño-Ballesteros, Singularities of Mappings The Local Behaviour of Smooth and Complex Analytic Mappings, 2020.
  * [Pfister-Popescu.75] G. Pfister, D. Popescu, Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math. 30 (1975), no. 2, 145–174.
  * [Popescu-Rond.19] D. Popescu, G. Rond, Remarks on Artin approximation with constraints. Osaka J. Math. 56 (2019), no. 3, 431–440.
  * [Rond.18] G. Rond, Artin approximation. J. Singul. 17 (2018), 108–192.
  * [Ruas] M. A. S. Ruas, On the finite $C^{l}$-determinacy and applications, PhD thesis, 1983, São-Carlos.
  * [Ruas.21] M. A. S. Ruas, Old and new results on density of stable mappings, to appear in Handbook of Singularities.II.
  * [Scherk.83] J. Scherk, À propos d’un théorème de Mather et Yau. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 12, 513–515
  * [Tougeron] J.C. Tougeron, Idéaux de fonctions différentiables. I. Ann. Inst. Fourier (Grenoble) 18 1968 fasc. 1, 177–240.
  * [Wall.81] C.T.C. Wall, Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13 (1981), no. 6, 481–539.
  * [Wall.95] C. T. C. Wall, Classification and stability of singularities of smooth maps. Singularity theory (Trieste, 1991), 920-952, World Sci. Publ., River Edge, NJ, 1995.
  * [Wieslaw] W. Wieslaw, Topological fields, M. Dekker (1988)


# Publication Trends in Astronomy: The Lone Author

Edwin A. Henneken

###### Abstract

In this short communication I highlight how the number of collaborators on
papers in the main astronomy journals has evolved over time. We see a trend of
moving away from single-author papers. This communication is based on data in
the holdings of the SAO/NASA Astrophysics Data System (ADS).

The ADS is funded by NASA Grant NNX09AB39G.

Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138

This communication illustrates the trend discussed by Mott Greene in the essay
“The demise of the lone author” (Greene (2007)). Trends are likely to be
different for different disciplines. As Mott observes: “In most fields outside
mathematics, fewer and fewer people know enough to work and write alone”. In
addition to this, in most disciplines large (and often multi-national)
collaborations have become more common and even unavoidable, because it is the
only way to get sufficient funding.

Figure 1 is an illustration of how the distribution of the number of authors
has changed over time in the main astronomy journals (The Astrophysical
Journal, The Astronomical Journal, Monthly Notices of the R.A.S. and Astronomy
& Astrophysics).

Figure 1.: The distribution of the relative frequency of the number of authors
per paper in the main astronomy journals for a number of years

Figure 2 highlights the “demise of the lone author” by showing the change in
the fraction of single author papers in the main astronomy journals. The
fraction in the main physics journals (Physical Review, Nuclear Physics,
Physics Letters) has been added for comparison.

Figure 2.: The fraction of papers by single authors in the main astronomy and
physics journals

The drop in the astronomy journals is more dramatic than for the physics
journals. A factor of about 10 versus a factor of about 3 or 4.

## References

  * Greene (2007) Greene, Mott. 2007, Nature, 450, 1165 (doi:10.1038/4501165a)