# On Tjurina Ideals of Hypersurface Singularities

João Hélder Olmedo Rodrigues

###### Abstract.

The Tjurina ideal of a germ of an holomorphic function $f$ is the ideal of
$\mathscr{O}_{\mathbbm{C}^{n},0}$ \- the ring of those germs at
$0\in\mathbbm{C}^{n}$ \- generated by $f$ itself and by its partial
derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the
structure of closed subscheme of $(\mathbbm{C}^{n},0)$ to the hypersurface
singularity defined by $f$, being an object of central interest in Singularity
Theory. In this note we introduce _$T$ -fullness_ and _$T$ -dependence_, two
easily verifiable properties for arbitrary ideals of germs of holomorphic
functions. These two properties allow us to give necessary and sufficient
conditions on an ideal $I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$, for the
equation $I=T(f)$ to admit a solution $f$.

Keywords: Hypersurface Singularities, Tjurina ideals, Analytic Classification

2010 Mathematics Subject Classification: 14B05, 14H20, 14D06, 32S05

The author was partially supported by FAPERJ, ARC 211.361/2019

##  1\. Introduction

Motivation for the present work comes from the analytic classification of
germs of hypersurfaces in $(\mathbbm{C}^{n},0)$. A germ of a complex
hypersurface $(X,0)\subset(\mathbbm{C}^{n},0)$ at the origin
$0\in\mathbbm{C}^{n}$ is defined as the zero set of some - non-trivial -
principal ideal $I(X)$ of $\mathscr{O}_{\mathbbm{C}^{n},0}$, the ring of germs
of holomorphic functions at $0\in\mathbbm{C}^{n}$. Let ${\mathfrak{m}}$ denote
the unique maximal ideal of $\mathscr{O}_{\mathbbm{C}^{n},0}$. A generator
$f\in{\mathfrak{m}}$ \- which is well-defined, modulo multiplication by an
invertible element in $\mathscr{O}_{\mathbbm{C}^{n},0}$ \- of $I(X)=(f)$ is
said to be an _equation_ for $(X,0)$; we often say in this case that
$(X,0)\subset(\mathbbm{C}^{n},0)$ _is defined by_ $f$.

As it is customary, we want to decide when two objects - germs of
hypersurfaces at $0\in\mathbbm{C}^{n}$, in the present case - are _alike_ , in
some prescribed sense. A quite natural notion comes from the _analytic
classification_ of germs of hypersurfaces. We say that two germs of
hypersurfaces, $(X_{1},0)\subset(\mathbbm{C}^{n},0)$ and
$(X_{2},0)\subset(\mathbbm{C}^{n},0)$ with equations $f_{1},\,f_{2}$ are
_analytically equivalent_ if there exist small open neighbourhoods $U_{1}$ and
$U_{2}$ of the origin $0\in\mathbbm{C}^{n}$, where $f_{1},\,f_{2}$ converge
and a (germ of) biholomorphism $\Phi:U_{1}\rightarrow U_{2}$ \- taking the
origin to itself - such that $\Phi(X_{1}\cap U_{1})=X_{2}\cap U_{2}$. This is
clearly an equivalence relation among germs of hypersurfaces. We refer to the
totality of germs of hypersurfaces all analytically equivalent to one another
as an _analytic class_. It is easy to see that $(X_{1},0)$ and $(X_{2},0)$ as
above are analytically equivalent if and only if there exists an automorphism
$\varphi$ of $\mathscr{O}_{\mathbbm{C}^{n},0}$ and an invertible element
$u\in\mathscr{O}_{\mathbbm{C}^{n},0}$ such that $\varphi(f_{1})=uf_{2}$. So, a
guiding big problem is to look for a complete set of invariants describing
completely analytic classes. For instance, one of the most famous invariants
of an analytic class is the _multiplicity_ of their elements. We recall that
if a system $\\{x_{1},\ldots,x_{n}\\}$ of generators of
${\mathfrak{m}}\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ is chosen and if
$(X,0)\subset(\mathbbm{C}^{n},0)$ is a germ of hypersurface defined by $f$,
its multiplicity ${\rm mult}(X,0)={\rm mult}(f)$ is the smallest degree $m$ of
a non-zero homogeneous polynomial appearing in a series expansion
$f(x_{1},\ldots,x_{n})=f_{m}(x_{1},\ldots,x_{n})+f_{m+1}(x_{1},\ldots,x_{n})+\ldots\,\,$
Clearly the multiplicity of a germ doesn’t depend on the choice of
$\\{x_{1},\ldots,x_{n}\\}$. So, two analytically equivalent germs of
hypersurfaces in $(\mathbbm{C}^{n},0)$ have the same multiplicity, but not the
other way around.

Given the previous terminology, the starting point for our research is a
result of J. Mather and S. Yau’s study [MY], which involves a certain
$\mathbbm{C}$-algebra associated to a germ of hypersurface
$(X,0)\subset(\mathbbm{C}^{n},0)$. Fix a system $\\{x_{1},\ldots,x_{n}\\}$ of
minimal generators of
${\mathfrak{m}}\subset\mathscr{O}_{\mathbbm{C}^{n},0}=\mathbbm{C}\\{x_{1},\ldots,x_{n}\\}$.
Suppose $(X,0)$ is defined by $f(x_{1},\ldots,x_{n})$. We define the _Tjurina
Algebra_ of $(X,0)$ \- or more accurately of $f$ \- as the quotient ring

$A(X)=A(f)=\mathbbm{C}\\{x_{1},\ldots,x_{n}\\}/(f,\frac{\partial f}{\partial
x_{1}},\ldots,\frac{\partial f}{\partial x_{n}}).$

The ideal $(f,\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial
f}{\partial x_{n}})$ appearing as the denominator is called the _Tjurina
ideal_ \- or extended Jacobian ideal - of $f$ and we denote it as $T(f)$. If
$g$ is another generator of $I(X)$ then $g=uf$, for some invertible element
$u\in\mathbbm{C}\\{x_{1},\ldots,x_{n}\\}$. Hence $A(f)=A(g)$, because actually
$T(f)=T(g)$ and this shows that $A(X)$ really doesn’t depend on the generator
of $I(X)$.

In order to explain Mather-Yau’s result - indeed an extension of it to the
case where the hypersurfaces involved are not required to have isolated
singularities, presented as Theorem 2.26 of [GLS] \- we observe that when
$(X_{1},0)$ and $(X_{2},0)$ are analytically equivalent germs of
hypersurfaces, then from a relation of type $\varphi(f_{1})=uf_{2}$ as above,
it is quite straightforward to check that the Tjurina algebras $A(X_{1})$ and
$A(X_{2})$ are isomorphic as $\mathbbm{C}$-algebras. The converse is far more
subtle and is (part of) the content of the aforementioned result, which we
state here in a simplified manner:

######  Theorem 1.1 (Mather, Yau).

_Two germs of complex hypersurfaces $(X_{1},0)\subset(\mathbbm{C}^{n},0)$ and
$(X_{2},0)\subset(\mathbbm{C}^{n},0)$ are analytically equivalent if and only
if $A(X_{1})$ and $A(X_{2})$ are isomorphic as $\mathbbm{C}$-algebras._

That’s why Mather and Yau baptised $A(X)$ as the _moduli algebra_ of the germ
of hypersurface $(X,0)\subset(\mathbbm{C}^{n},0)$: the preceding result tells
us that the problem of the classification of germs of hypersurfaces
$(X,0)\subset(\mathbbm{C}^{n},0)$ up to analytic equivalence is equivalent to
that of the classification of their Tjurina algebras $A(X)$ up to
$\mathbbm{C}$-algebra isomorphism.

In this vein, and with an eye in the big analytic problem of classification of
germs of hypersurfaces $(X,0)\subset(\mathbbm{C}^{n},0)$ mentioned above, we
ask ourselves what seems to be a prior - algebraic, of course - question,
namely:

######  Question 1.2.

Fixed $n\geqslant 1$, how to find necessary and sufficient conditions for an
ideal $I\,\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ to be the Tjurina ideal of
some $f\in{\mathfrak{m}}$?

Clearly, the zero ideal and the whole ring $\mathscr{O}_{\mathbbm{C}^{n},0}$
are Tjurina ideals of $0$ and $x_{1}$, say, respectively.

######  Example 1.3.

If $n=1$ then any ideal
$I\subset\mathscr{O}_{\mathbbm{C}^{1},0}=\mathbbm{C}\\{x_{1}\\}$ is of the
form $I={\mathfrak{m}}^{k}=(x_{1}^{k})$, $k\geqslant 0$; so it is the Tjurina
ideal of $f(x_{1})=x_{1}^{k+1}$.

So, if $n=1$, Question 1.2 has a trivial answer: any ideal is the Tjurina
ideal of some function. However, beginning at $n=2$, this is not true any
more. Clearly a necessary condition for an ideal
$I\,\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ to be the Tjurina ideal of some
$f\in\mathscr{O}_{\mathbbm{C}^{n},0}$ is that the minimal number of generators
of $I$ should be at most $n+1$, but this is not sufficient as we see in the
next

######  Example 1.4.

Let
$I={\mathfrak{m}}^{2}=(x_{1}^{2},x_{1}x_{2},x_{2}^{2})\subset\mathbbm{C}\\{x_{1},x_{2}\\}=\mathscr{O}_{\mathbbm{C}^{2},0}$
\- i.e. a planar triple point - is not of the form $T(f)$, for any
$f(x_{1},x_{2})\in\mathscr{O}_{\mathbbm{C}^{2},0}$. Indeed,
$\dim_{\mathbbm{C}}\,\Big{(}\frac{\mathscr{O}_{\mathbbm{C}^{2},0}}{I}\Big{)}=3$
and it is known (cf. [H1], Example 14.1) that there exists - up to analytic
change of coordinates - only one $f(x_{1},x_{2})$ such that
$\dim_{\mathbbm{C}}\,\Big{(}\frac{\mathscr{O}_{\mathbbm{C}^{2},0}}{T(f)}\Big{)}=3$,
namely $x_{1}^{4}+x_{2}^{2}$. Now in
$\frac{\mathscr{O}_{\mathbbm{C}^{2},0}}{I}$ any non-invertible $\xi$ is such
that $\xi^{2}=0$, while in
$\frac{\mathscr{O}_{\mathbbm{C}^{2},0}}{T(x_{1}^{4}+x_{2}^{2})}$,
$\eta=\bar{x_{1}}$, the residual class of $x_{1}$, is such that $\eta^{2}\neq
0$.

######  Example 1.5.

Let $n\geqslant 2$ and consider
$I=(x_{1}x_{2},\ldots,x_{n-1}x_{n})=(\\{x_{i}x_{j}\\}_{i<j})$, an ideal
defining the union of $n$ lines in $(\mathbbm{C}^{n},0)$ meeting transversally
at $0$. If $n\geqslant 4$, then $I$ is not a Tjurina ideal of any
$f\in\mathscr{O}_{\mathbbm{C}^{n},0}$. Indeed, $I$ is minimally generated by
$\frac{n(n-1)}{2}$ monomials, which exceeds $n+1$ if $n\geqslant 4$. If $n=3$
we see that for $f=x_{1}x_{2}x_{3}$,
$T(f)=(x_{2}x_{3},x_{1}x_{3},x_{1}x_{2})=I$. We will show below (Proposition
3.3) that in the case $n=2$, $I=(x_{1}x_{2})$ is _not_ a Tjurina ideal.

Notice that for a given
$f\in{\mathfrak{m}}\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ the ideal $T(f)$
defines the _singular (closed) subscheme_ of the germ
$(X,0)\subset(\mathbbm{C}^{n},0)$ defined by $f$. So our Question 1.2 above
has an algebraic-geometric counterpart: which closed subchemes of
$(\mathbbm{C}^{n},0)$ are singular subschemes of germs of hypersurfaces in
$(\mathbbm{C}^{n},0)$? With this geometric viewpoint it is of course important
to emphasize that the closed subscheme of $(\mathbbm{C}^{n},0)$ defined by
some ideal $I$ is in general non-reduced and shall not be confused with the
one defined by the radical $\sqrt{I}$, which is reduced. For example, in
$(\mathbbm{C}^{2},0)$ the ideal ${\mathfrak{m}}$ is the Tjurina ideal of
$f=x_{1}x_{2}$ but the previous Example 1.4 has shown that the ideal
${\mathfrak{m}}^{2}$ is not a Tjurina ideal.

With this scheme theoretic phrasing, our question was addressed previously by
Aluffi in [A] using what he refers to as the _$\mu$ -class_ of the closed
subscheme, in an appropriate Chow group. Aluffi, in (loc.cit.) obtains
obstructions for a closed scheme to be the singular scheme of some
hypersurface and out of his considerations, several interesting _non-examples_
\- subschemes that are _not_ singular subschemes of hyperfurfaces - are
presented. The $\mu$-class is defined only for singular schemes of
hypersurfaces. Our approach works for _any_ closed subscheme of
$(\mathbbm{C}^{n},0)$ (or rather any ideal of
$\mathscr{O}_{\mathbbm{C}^{n},0}$) providing sufficient and necessary
conditions for an ideal $I$ to be a Tjurina ideal.

We now describe the contents of the paper. In Section 2, along with some
preliminaries on Tjurina ideals, we introduce and investigate some properties
of the set $\Delta(I)$ of antiderivatives of an arbitrary ideal $I$, which is
a natural set of elements of $\mathscr{O}_{\mathbbm{C}^{n},0}$ to look at in
the search for a solution $f$ for the equation $I=T(f)$. We also suggest an
easily appliable method for computation of $\Delta(I)$ in examples, with
routines already implemented in _SINGULAR_ [DGPS]. In Section 3 we present, by
means of the introduction of two simple and quite computable, operations on
arbitrary ideals of $\mathscr{O}_{\mathbbm{C}^{n},0}$, our main result, namely
a characterization of the ideals $I$ for which the equation $I=T(f)$ admits a
solution $f$. In other words, we characterize Tjurina ideals giving an
explicit answer for Question 1.2. In Section 4, some properties of the
formation of Tjurina ideals are discussed.

##  2\. Generalities

###  2.1. Tjurina Ideals

As we have indicated in the Introduction, we will be concerned with Tjurina
ideals of germs of functions $f\in\mathscr{O}_{\mathbbm{C}^{n},0}$. In what
follows we fix, once and for all, a basis $\\{x_{1},\ldots,x_{n}\\}$ \-
minimal set of generators - of the maximal ideal
${\mathfrak{m}}\subset\mathscr{O}_{\mathbbm{C}^{n},0}$; then we identify
$\mathscr{O}_{\mathbbm{C}^{n},0}=\mathbbm{C}\\{x_{1},\ldots,x_{n}\\}$ and
compute $T(f)=(f,\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial
f}{\partial x_{n}})$ with respect to this basis. Occasionally, in examples
with small $n$ we use $x,\,\,\\{x,y\\},\,\,\\{x,y,z\\},$ etc. instead of
$x_{1},\,\,\\{x_{1},x_{2}\\},\,\,\\{x_{1},x_{2},x_{3}\\},$ etc. Below we
collect some elementary, straightforward properties of the formation of
Tjurina ideals:

######  Remark 2.1.

Let $f,\,g\in\mathscr{O}_{\mathbbm{C}^{n},0}$. Then

$T(f+g)\subseteq T(f)+T(g)$;

$T(fg)\subseteq fT(g)+gT(f)\subseteq T(f)T(g)\subseteq T(f)\cap T(g)$.

If $(f)=(g)$ then $T(f)=T(g)$;

Notice that $(iii)$ in the previous Remark allows us to talk about $T$ _of a
principal ideal_. It will be useful to be able to extend $T$ to an arbitrary
ideal, as follows.

######  Definition 2.2.

Let $J=(g_{1},\ldots,g_{q})$ be an ideal of $\mathscr{O}_{\mathbbm{C}^{n},0}$.
We denote $T(J)$ the ideal $T(g_{1})+\ldots+T(g_{q})$.

It is an easy matter, based on general properties (cf. Remark 2.1), to check
that indeed the ideal $T(J)$ doesn’t depend on the particular choice of
generators for $J$. It shall be referred to as _the Tjurina ideal of $J$_. The
properties below are easy to verify.

######  Remark 2.3.

$J\subseteq T(J)$;

If $J_{1}\subseteq J_{2}$ is an inclusion of ideals then $T(J_{1})\subseteq
T(J_{2})$;

If $\\{J_{\lambda}\\}_{\lambda\in\Lambda}$ is any family of ideals of
$\mathscr{O}_{\mathbbm{C}^{n},0}$ then
$T(\sum_{\lambda}J_{\lambda})=\sum_{\lambda}T(J_{\lambda})$;

$T(J_{1}\cap J_{2})\subseteq T(J_{1})\cap T(J_{2})$.

In addition to the preceding considerations relative to Tjurina ideals and in
view of our guiding Question 1.2 in which an ideal $I$ is fixed, it is natural
to introduce the concept below:

######  Definition 2.4.

For any given ideal $I\subseteq\mathscr{O}_{\mathbbm{C}^{n},0}$ we define the
_set of antiderivatives_ of $I$ by means of

$\Delta(I)=\\{f\in\mathscr{O}_{\mathbbm{C}^{n},0}\,|\,T(f)\subseteq I\\}.$

An expression for $\Delta(I)$, free of coordinates, can be given. We recall
that a $\mathbbm{C}$-derivation $\delta$ of $\mathscr{O}_{\mathbbm{C}^{n},0}$
is a $\mathbbm{C}$-linear map $\delta$ from $\mathscr{O}_{\mathbbm{C}^{n},0}$
to itself that satisfies the Leibniz’ rule: for all
$f,\,g\in\mathscr{O}_{\mathbbm{C}^{n},0}$, $\delta(fg)=f\delta(g)+g\delta(f).$
The set $Der_{\mathbbm{C}}(\mathscr{O}_{\mathbbm{C}^{n},0})$ of
$\mathbbm{C}$-derivations of $\mathscr{O}_{\mathbbm{C}^{n},0}$ is naturally a
$\mathscr{O}_{\mathbbm{C}^{n},0}$-module, which is freely generated by the
partial derivatives $\frac{\partial}{\partial x_{i}}$, $i=1,\ldots,n$. It is
then easy to check that $\Delta(I)$ coincides with $\\{f\in I\,|\,\delta(f)\in
I,\,\mbox{for
every}\,\,\delta\in\,Der_{\mathbbm{C}}(\mathscr{O}_{\mathbbm{C}^{n},0})\,\\}.$
This shows in particular that $\Delta(I)$ does not depend on our choice of
parameters $\\{x_{1},\ldots,x_{n}\\}$ used to compute Tjurina ideals.

If $I$ is a non-zero ideal of $\mathscr{O}_{\mathbbm{C}^{n},0}$, we use
$\mathop{\rm ord}\nolimits(I)$ to denote $\operatorname{min}\\{\mathop{\rm
ord}\nolimits(g)\,|\,g\in
I\\}=\operatorname{max}\\{k\,|\,I\subseteq{\mathfrak{m}}^{k}\\}$. Below we
summarize some easy properties of $\Delta$.

######  Remark 2.5.

$\Delta(I)$ is an ideal of $\mathscr{O}_{\mathbbm{C}^{n},0}$;

$I^{2}\subseteq\Delta(I)\subseteq I$;

If $\\{I_{\lambda}\\}_{\lambda\in\Lambda}$ is any family of ideals of
$\mathscr{O}_{\mathbbm{C}^{n},0}$ then
$\Delta(\bigcap_{\lambda}I_{\lambda})=\bigcap_{\lambda}\Delta(I_{\lambda})$;

If $I\subseteq J$ is an inclusion of ideals in
$\mathscr{O}_{\mathbbm{C}^{n},0}$ then $\Delta(I)\subseteq\Delta(J)$;

If $I\neq(0)$ and $I\neq(1)$ then $\mathop{\rm
ord}\nolimits(\Delta(I))\geqslant\mathop{\rm ord}\nolimits(I)+1$.

###### Proof.

The verification of $(i)$ to $(iv)$ is straightforward. When it comes to
$(v)$, we always have $\mathop{\rm
ord}\nolimits(\Delta(I))\geqslant\mathop{\rm ord}\nolimits(I)$. If equality
holds, take $g\in\Delta(I)$ such that $\mathop{\rm
ord}\nolimits(g)=\mathop{\rm ord}\nolimits(\Delta(I))=\mathop{\rm
ord}\nolimits(I)$. In this case all partial derivatives $\frac{\partial
g}{\partial x_{i}}$ of $g$ would belong to $I$ and we would obtain
$\mathop{\rm ord}\nolimits(\frac{\partial g}{\partial
x_{i}})\geqslant\mathop{\rm ord}\nolimits(I)=\mathop{\rm ord}\nolimits(g)$.
This is only possible if $g$ is an invertible element of
$\mathscr{O}_{\mathbbm{C}^{n},0}$. Hence $I=(1)$. ∎

######  Example 2.6.

Let ${\mathfrak{m}}^{k}$, with $k\geqslant 1$, denote the $k$-th power of the
maximal ideal of $\mathscr{O}_{\mathbbm{C}^{n},0}$. Clearly
$\Delta({\mathfrak{m}}^{k})\supset{\mathfrak{m}}^{k+1}$ and according to
Remark 2.5, (v), if $f\in\Delta({\mathfrak{m}}^{k})$ then $\mathop{\rm
ord}\nolimits(f)\geqslant\mathop{\rm ord}\nolimits({\mathfrak{m}}^{k})+1=k+1$.
Hence, $f\in{\mathfrak{m}}^{k+1}$. It follows that
$\Delta({\mathfrak{m}}^{k})={\mathfrak{m}}^{k+1}$.

######  Example 2.7.

Let $I=(f^{k})$ be a principal ideal, $k\geqslant 1$, with
$f\in\mathscr{O}_{\mathbbm{C}^{n},0}$ an irreducible element. Then
$\Delta(I)=(f^{k+1})$: because if $g\in\Delta(I)$, we have $g=af^{k}$, for
some $a\in\mathscr{O}_{\mathbbm{C}^{n},0}$ and then, for all $i=1\ldots,n$,
$\frac{\partial g}{\partial x_{i}}=f^{k}.\frac{\partial a}{\partial
x_{i}}+akf^{k-1}\frac{\partial f}{\partial x_{i}}$ is also a multiple of
$f^{k}$. Hence $f$ must divide $ka\frac{\partial f}{\partial x_{i}}$, for all
$i=1\ldots,n$. Now, if $f$ divides $\frac{\partial f}{\partial x_{i}}$ for all
$i=1\ldots,n$ then $(\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial
f}{\partial x_{n}})\subseteq(f)$ and we would deduce as in [B], Lemma 1.2.13,
that $f=0\in\mathscr{O}_{\mathbbm{C}^{n},0}$, which is not the case. So, for
some $i$, $f$ does not divide $\frac{\partial f}{\partial x_{i}}$ and, being
an irreducible element, $f$ must divide $a$. Hence $g\in(f^{k+1})$.

######  Remark 2.8.

Notice that if one is interested in fields $\mathbbm{k}$ of positive
characteristic $p$, the preceding two Examples are false, being enough to take
$I=(x_{1}^{p})\subset\mathbbm{k}[[x_{1}]]$ as counterexamples. Indeed, in this
case $\Delta(I)=I\neq(x_{1}^{p+1})$.

######  Proposition 2.9.

If $\mathfrak{p}\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ is a prime ideal and
$I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ is a $\mathfrak{p}$-primary ideal,
then its ideal of antiderivatives $\Delta(I)$ is also a $\mathfrak{p}$-primary
ideal.

###### Proof.

The $\mathfrak{p}$-primary part of the statement is consequence of Remark 2.5,
(ii). So, we need only to check that $\Delta(I)$ is primary, that is, to show
that if $f,g\in\mathscr{O}_{\mathbbm{C}^{n},0}$, $fg\in\Delta(I)$ and
$g\not\in\Delta(I)$, then $f\in\sqrt{\Delta(I)}$. We argue as follows: if
$g\not\in\Delta(I)$ there are two possibilities to consider. The first one is
$g\not\in I$. In this case, as $I$ is primary, and $fg\in I$ there exists some
$k\geqslant 1$ such that $f^{k}\in I$. The second possibility is that $g\in I$
and then it is forced that, for some $i=1,\ldots,n$, $\frac{\partial
g}{\partial x_{i}}\not\in I$. From $\frac{\partial(fg)}{\partial x_{i}}\in I$
we deduce $f\frac{\partial g}{\partial x_{i}}\in I$. Again using that $I$ is
primary, there exists some $k\geqslant 1$ such that $f^{k}\in I$. So we see
that in any of the two possibilities, $f^{k}\in I$. It is then easy to check
that $f^{k+1}\in\Delta(I)$ as we wanted ∎

We observe that known theorems (see Theorem 7.13 in [AM], for example) about
primary decomposition of ideals in noetherian rings, Remark 2.5,(iv) and the
previous result imply that when it comes to compute $\Delta(I)$ it is possible
to restrict attention to primary ideals. One instance where this observation
can be applied directly is presented in the example below.

######  Example 2.10.

Let $I=(f)$ be a principal ideal and $f=f_{1}^{k_{1}}\ldots f_{r}^{k_{r}}$ be
a factorization of $f$ into irreducible, non-associated, elements with
$k_{1},\ldots,k_{r}\geqslant 1$. Then
$I=(f_{1}^{k_{1}})\cap\ldots\cap(f_{r}^{k_{r}})$. It follows from Remark 2.5,
(iii) and Example 2.7 that
$\Delta(I)=(f_{1}^{k_{1}+1})\cap\ldots\cap(f_{r}^{k_{r}+1})=(f_{1}^{k_{1}+1}\cdot\ldots\cdot
f_{r}^{k_{r}+1})$.

###  2.2. Computation of $\Delta(I)$

To this point we have computed explicitly the ideal of antiderivatives
$\Delta(I)$ only in the case where $I$ is a power of the maximal ideal (cf.
Example 2.6) or $I$ is a principal ideal (cf. Example 2.10). Now we suggest a
method to compute $\Delta(I)$ for an arbitrary ideal
$I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$. Our procedure will be illustrated
with a number of examples obtained using basic routines already implemented in
the software _SINGULAR_ , [DGPS].

Fix a basis $\\{x_{1},\ldots,x_{n}\\}$ \- minimal set of generators - of the
maximal ideal ${\mathfrak{m}}\subset\mathscr{O}_{\mathbbm{C}^{n},0}$; we
identify $\mathscr{O}_{\mathbbm{C}^{n},0}=\mathbbm{C}\\{x_{1},\ldots,x_{n}\\}$
as before. A free $\mathscr{O}_{\mathbbm{C}^{n},0}$-module of certain rank
$\ell$ will be denoted by $F_{\ell}$; also, for any
$\underline{b}=(b_{1},\ldots,b_{\ell})^{t}$,
$\underline{c}=(c_{1},\ldots,c_{\ell})^{t}\in F_{\ell}$ we put
$\langle\underline{b}\,,\underline{c}\rangle$ to denote the element
$\sum_{k=1}^{\ell}b_{k}c_{k}\in\mathscr{O}_{\mathbbm{C}^{n},0}$.

We assume the ideal $I$ given by generators $I=(f_{1},\ldots,f_{s})$. Then
some element
$g=\langle\underline{a}\,,\underline{f}\rangle=\sum_{k}a_{k}f_{k}\in I$ is an
element of $\Delta(I)$ if and only if, for all $j=1,\ldots,n$, the partial
derivative

$\frac{\partial g}{\partial x_{j}}=\sum_{k}a_{k}\frac{\partial f_{k}}{\partial
x_{j}}+\sum_{k}\frac{\partial a_{k}}{\partial
x_{j}}f_{k}=\big{\langle}\underline{a}\,,\frac{\partial\underline{f}}{\partial
x_{j}}\big{\rangle}+\big{\langle}\frac{\partial\underline{a}}{\partial
x_{j}}\,,\underline{f}\big{\rangle}$

belongs to $I$. In other words,
$g=\langle\underline{a}\,,\underline{f}\rangle\in I$ if and only if
$\big{\langle}\underline{a}\,,\frac{\partial\underline{f}}{\partial
x_{j}}\big{\rangle}\in I,$ for all $j=1,\ldots,n$.

For each $j=1,\ldots,n$ we denote by $M^{j}$ the submodule of $F_{s}$
consisting of elements $\underline{a}\in F_{s}$ such that
$\big{\langle}\underline{a}\,,\frac{\partial\underline{f}}{\partial
x_{j}}\big{\rangle}\in I$. Then $M:=\bigcap_{j=1}^{n}\,M^{j}\subseteq F_{s}$
is a finitely generated $\mathscr{O}_{\mathbbm{C}^{n},0}$-submodule of $F_{s}$
consisting of elements $\underline{a}\in\,F_{s}$ such that
$g=\langle\underline{a}\,,\underline{f}\rangle\in\Delta(I)$. Let
$\underline{m}_{\,1},\ldots,\underline{m}_{\,q}\in F_{s}$ be generators of
$M$: $\underline{m}_{\,i}=(m_{1i},\ldots,m_{si})^{t}$, where
$m_{ki}\in\mathscr{O}_{\mathbbm{C}^{n},0}$.

Notation being as above we have

######  Proposition 2.11.

The ideal of antiderivatives $\Delta(I)$ is generated by
$\langle\underline{m}_{\,i}\,,\underline{f}\rangle$, for $i=1,\ldots,q$.

###### Proof.

Because $g\in\Delta(I)$ if and only if there exists some $\underline{a}\in M$
such that $g=\langle\underline{a}\,,\underline{f}\rangle$. Since $M$ is
generated by the $\underline{m}_{\,i}$, the result follows from the
$\mathscr{O}_{\mathbbm{C}^{n},0}$-linearity of
$\langle{-}\,,\underline{f}\rangle$. ∎

Here we illustrate how we used the software _SINGULAR_ , [DGPS] to compute the
ideal of antiderivatives of a given ideal. We do not claim originality, since
we only applied routines already implemented by the software developers and
collaborators.

######  Example 2.12.

Let
$I=(x^{2},xy,yz,z^{2},y^{2}-xz)\subset\mathscr{O}_{\mathbbm{C}^{3},0}=\mathbbm{C}\\{x,y,z\\}$.
With the interface of SINGULAR open, type

LIB "hnoether.lib";
ring r=0,(x,y,z),ds;

This declares you are working over a field of characteristic zero (the field
of rational numbers, indeed), variables $x,y,z$ and says we work over the
corresponding ring of power series. We now declare the generators of the ideal
$I$ by means of a matrix with one row and (in the present case) five columns:
type

matrix B[1][5]=x2,xy,yz,z2,y2-xz;

Now to compute the submodule $M^{1}$, (same notation as above), we declare a
matrix with the partial derivatives of the given generators in terms of $x$ as
entries. Then compute $M^{1}$ as follows:

matrix A1[1][5]=2x,y,0,0,-z; def m1=modulo(A1,B);

Likewise, compute $M^{2}$ and $M^{3}$ by taking partial derivatives with
respect to $y$ and $z$:

matrix A2[1][5]=0,x,z,0,2y; def m2=modulo(A2,B);
matrix A3[1][5]=0,0,y,2z,-x; def m3=modulo(A3,B);

Now, we put

def m=intersect(m1,m2,m3);
def M=std(m);

This is the definition of $M$ and the computation of a standard basis, with
respect to the given monomial order. Now, to obtain $M$, type

print(M);

In this case _SINGULAR_ , [DGPS] gives

$\left(\begin{array}[]{ccccccccccccccc}z&0&0&-y&x&0&0&0&-z&y&0&0&0&0&z^{2}\\\
0&0&-z&2x&0&-2z&0&0&2y&0&0&0&0&z^{2}&0\\\ 0&-2y&3x&0&0&0&0&y&0&0&0&z&0&0&0\\\
0&x&0&0&0&0&y&0&0&0&z&0&0&0&0\\\ 2x&0&-y&0&0&y&0&z&0&0&0&0&z^{2}&0&0\\\
\end{array}\right)$.

Hence, as in the previous Proposition, we compute generators of $\Delta(I)$
computing

$\langle\,\cdot\,,(x^{2},xy,yz,z^{2},y^{2}-xz)\rangle,$

for all (transposed) columns of the previous matrix. We obtain (after cleaning
up superfluous generators)

$\Delta(I)=(x^{3},x^{2}y,2xy^{2}-x^{2}z,\,y^{3}-3xyz,\,2y^{2}z-xz^{2},\,yz^{2},z^{3},x^{2}z^{2}).$

##  3\. Two key conditions

###  3.1. T-fullness

We now introduce $T$-fullness, a natural concept which together with
$T$-dependence (cf. Definition 3.7 below) is present in our characterization
of Tjurina ideals of $\mathscr{O}_{\mathbbm{C}^{n},0}$ (see Theorem 3.12).

Recall the definition of the Tjurina ideal of an arbitrary ideal of
$\mathscr{O}_{\mathbbm{C}^{n},0}$ (cf. Definition 2.2).

######  Definition 3.1.

Let $I$ be an ideal of $\mathscr{O}_{\mathbbm{C}^{n},0}$. In general we have
$T(\Delta(I))\subseteq I$. We say that _$I$ is $T$-full_ if $I=T(\Delta(I))$.
.

######  Example 3.2.

Let
$I=(xy,x^{4}+y^{3})\subset\mathscr{O}_{\mathbbm{C}^{2},0}=\mathbbm{C}\\{x,y\\}$.
Following the routine of SINGULAR, [DGPS], described as in the example used to
illustrate Proposition 2.11, we compute

$M=\begin{pmatrix}4y^{2}&3x^{3}&xy&y^{3}&x^{4}\\\ x&y&0&0&0\\\
\end{pmatrix}\subset F_{2}.$

Hence, $\Delta(I)=(5xy^{3}+x^{5},4x^{4}y+y^{4},x^{2}y^{2},xy^{4},x^{5}y)$.
Then $xy\not\in(x^{2}y,xy^{2},x^{4}+y^{3},x^{5})=T(\Delta(I))$, so $I$ is not
$T$-full.

$T$-fullness is a property that characterizes Tjurina ideals in the special
case of principal ideals. This is discussed in detail in the result below:

######  Proposition 3.3.

Let $n\geqslant 2$ and $I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ be a (non-
trivial) principal ideal.

  * (i)

If $I=(f^{k})$ with $f\in\mathscr{O}_{\mathbbm{C}^{n},0}$ irreducible,
$k\geqslant 1$. Then $I$ is a Tjurina ideal if and only if $f$ has
multiplicity one;

  * (ii)

Let $I=(f)$, $f=f_{1}^{k_{1}}\ldots f_{r}^{k_{r}}$, with non associated
irreducible factors $f_{i}$ and $r\geqslant 2$. Then $I$ is not a Tjurina
ideal;

  * (iii)

$I$ is a Tjurina ideal if and only if $I$ is $T$-full.

###### Proof.

If $I=(f^{k})$ with $f$ irreducible then as we have seen in Example 2.7,
$\Delta(I)$ consists of multiples of $f^{k+1}$. Hence
$T(\Delta(I))=T(f^{k+1})=f^{k}T(f)=IT(f)$. By Nakayama’s Lemma we conclude
that $I$ is a Tjurina ideal if and only if
$T(f)=(1)=\mathscr{O}_{\mathbbm{C}^{n},0}$. Since $f\in{\mathfrak{m}}$ this
last condition occurs if and only if $f$ has multiplicity one. This is part
$(i)$.

Now, let $I=(f)$ be a principal ideal and $f=f_{1}^{k_{1}}\ldots
f_{r}^{k_{r}}$ as in $(ii)$. We have seen in Example 2.10 that $\Delta(I)$
consists of multiples of $f_{1}^{k_{1}+1}\cdot\ldots\cdot f_{r}^{k_{r}+1}$.
Hence $T(\Delta(I))=T(f_{1}^{k_{1}+1}\cdot\ldots\cdot f_{r}^{k_{r}+1})$. A
routine computation using Leibniz’ rule shows that for all $j=1,\ldots,n$,

$\frac{\partial(f_{1}^{k_{1}+1}\cdot\ldots\cdot f_{r}^{k_{r}+1})}{\partial
x_{j}}\in(f_{1}^{k_{1}}\cdot\ldots\cdot f_{r}^{k_{r}})\cdot(\\{f_{1}\ldots
f_{i-1}\frac{\partial f_{i}}{\partial x_{j}}f_{i+1}\ldots
f_{r}\\}_{i=1}^{r}).$

As a result $T(f_{1}^{k_{1}+1}\cdot\ldots\cdot f_{r}^{k_{r}+1})\subseteq IK$,
where $K$ is the ideal generated by the $f_{1}\cdot\ldots\cdot f_{r}$ and

$\\{f_{1}\ldots f_{i-1}\frac{\partial f_{i}}{\partial x_{j}}f_{i+1}\ldots
f_{r}\\}_{i,j},$

where $i\in\\{1,\ldots,r\\}$ and $j\in\\{1,\ldots,n\\}$. Hence
$T(\Delta(I))\subseteq{\mathfrak{m}}^{r-1}I$ and since $r\geqslant 2$ we
conclude, using Nakayama’s Lemma, that $I$ is not a Tjurina ideal. This
finishes part $(ii)$.

Part $(iii)$ is a consequence of the proofs presented for $(i)$ and $(ii)$. ∎

######  Example 3.4.

The previous Proposition 3.3, $(i)$ shows that there is no $f$ with $T(f)$
equal to, say, $I=(x_{1}^{2}+x_{2}^{3})$. Besides, $(ii)$ applies for
$I=(x_{1}x_{2})$, (the case $n=2$ of Example 1.5) showing that $I$ is not a
Tjurina ideal.

The general significance (beyond principal ideals) of $T$-fullness in our
investigation is apparent in the next result.

######  Proposition 3.5.

Let $I$ be an ideal $\mathscr{O}_{\mathbbm{C}^{n},0}$. If $I$ is a Tjurina
ideal then $I$ is $T$-full.

###### Proof.

Let $\Delta(I)=(g_{1},\ldots,g_{q})$. If $I$ is a Tjurina ideal, then there
exists $f\in\Delta(I)$ such that $I=T(f)$. We may write
$f=\sum_{i=1}^{q}r_{i}g_{i}$, for some
$r_{i}\in\mathscr{O}_{\mathbbm{C}^{n},0}$. Using Remark 2.1, we obtain
$I=T(f)\subseteq\sum_{i=1}^{q}T(r_{i}g_{i})\subseteq\sum_{i=1}^{q}T(g_{i})=T\Delta(I)\subseteq
I$ and equality holds throughout. We conclude that $I$ is $T$-full. ∎

In the general case $T$-fullness is not sufficient for $I$ to be a Tjurina
ideal as shows the next

######  Example 3.6.

Let
$I=(x^{2},xy,y^{2})={\mathfrak{m}}^{2}\subset\mathscr{O}_{\mathbbm{C}^{2},0}=\mathbbm{C}\\{x,y\\}$.
We have
$\Delta({\mathfrak{m}}^{2})={\mathfrak{m}}^{3}=(x^{3},x^{2}y,xy^{2},y^{3})$,
so $T(\Delta(I))=T(x^{3})+T(x^{2}y)+T(xy^{2})+T(y^{3})={\mathfrak{m}}^{2}=I$
and $I$ is $T$-full. We have seen before that $I$ is not a Tjurina ideal.

###  3.2. T-dependence

Here we explain the last ingredient which is, together with $T$-fullness (cf.
Definition 3.1), present in our characterization of Tjurina ideals.

We begin with an example. Consider the ideal $I={\mathfrak{m}}$ and its ideal
of antiderivatives $\Delta(I)={\mathfrak{m}}^{2}=(x^{2},xy,y^{2})$ inside
$\mathscr{O}_{\mathbbm{C}^{2},0}=\mathbbm{C}\\{x,y\\}$. We notice that a
_general_ $\mathbbm{C}$-linear combination of the given generators of
$\Delta(I)$ is a germ $f$ satisfying $I=T(f)$. By contrast, for _special_
linear combinations of the generators, the corresponding $T(f)$ is smaller
than $I$.

We aim to coin a concept which encompasses the above example and systematizes
the intuition, that of a general linear combination $f$ of the generators of
$\Delta(I)$ has the largest $T(f)$ and could reveal whether a given ideal $I$
is a Tjurina ideal. Our results and definitions here are expressed more in a
geometric language, which seemed to us appropriate. So, we use standard
concepts on schemes, consistent with [H2], Chapter II, to which we refer for
unexplained terminology.

Let $J\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ be any ideal. We regard $J$ as
an ideal sheaf on $\mbox{Spec}\,\mathscr{O}_{\mathbbm{C}^{n},0}$. Suppose $J$
is given by generators:
$J=(g_{1},\ldots,g_{q})\subset\mathscr{O}_{\mathbbm{C}^{n},0}$; then consider
the projective $(q-1)$-space over $\mathscr{O}_{\mathbbm{C}^{n},0}$, namely,
${\mathbbm{P}}={\mathbbm{P}}^{q-1}=\mbox{Proj}(S)$, where
$S=\mathscr{O}_{\mathbbm{C}^{n},0}[\alpha_{1},\ldots,\alpha_{q}]=\bigoplus_{d\geqslant
0}\,S_{d}$ is the ring of polynomials over $\mathscr{O}_{\mathbbm{C}^{n},0}$,
with variables $\alpha_{i}$, graded so that $\operatorname{deg}\alpha_{i}=1$,
for all $i$.

Let
$\pi:{\mathbbm{P}}^{q-1}\rightarrow\mbox{Spec}\,\mathscr{O}_{\mathbbm{C}^{n},0}$
be the natural morphism of
$\mbox{Spec}\,\mathscr{O}_{\mathbbm{C}^{n},0}$-schemes and let $\sigma$ denote
the global section $\sum_{i=1}^{q}\,g_{i}\alpha_{i}$ of
$\pi^{*}(J)\otimes\mathscr{O}_{{\mathbbm{P}}}(1)$. Then there is a _Tjurina
ideal sheaf_ of $\sigma$ on ${\mathbbm{P}}^{q-1}$, namely
$\mathscr{T}(\sigma)$, the sheaf associated to the homogeneous ideal
$(\sigma,\frac{\partial\sigma}{\partial
x_{1}},\ldots,\frac{\partial\sigma}{\partial x_{n}})$ of $S$. Clearly
$\mathscr{T}(\sigma)$ is a subsheaf of $\pi^{*}(T(J))$, so we can consider the
quotient sheaf $\mathscr{F}=\pi^{*}(T(J))/\mathscr{T}(\sigma)$ on
${\mathbbm{P}}^{q-1}$. For later use, we observe that since $\mathscr{F}$ is
coherent and ${\mathbbm{P}}^{q-1}$ is a noetherian scheme, the support
$\mbox{Supp}\mathscr{F}$ of $\mathscr{F}$ is a closed subscheme of
${\mathbbm{P}}^{q-1}$ defined by the vanishing of
$(\mathscr{T}(\sigma):\pi^{*}(T(J))$.

######  Definition 3.7.

We say that an ideal $J\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ is _$T$
-dependent_ if $\pi^{-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F}$.
Equivalently, $J$ is $T$-dependent if and only if
$(\mathscr{T}(\sigma):\pi^{*}(T(J))\not\subseteq{\mathfrak{m}}S$.

We check that $T$-dependence for $J$ is a well-defined concept, being
independent on the choice of generators for $J$. For, let the ideal $J$ be
given by another system of generators, say $J=(h_{1},\ldots,h_{u})$. Let
${\mathbbm{P}}^{u-1}=\mbox{Proj}(S^{\prime})$, being
$S^{\prime}=\mathscr{O}_{\mathbbm{C}^{n},0}[\,\beta_{1},\ldots,\beta_{u}]$.
Carry the above construction with the obvious morphism
$\pi^{\prime}:{\mathbbm{P}}^{u-1}\rightarrow\mbox{Spec}\,\mathscr{O}_{\mathbbm{C}^{n},0}$
and the section $\sigma^{\prime}=\sum_{j=1}^{u}\,h_{j}\beta_{j}$ instead of
$\pi$ and $\sigma$, obtaining a sheaf $\mathscr{F^{\prime}}$ on
${\mathbbm{P}}^{u-1}$.

######  Lemma 3.8.

Notation being as above,
$\pi^{-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F}$ if and only if
$\pi^{\prime-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F^{\prime}}$.

###### Proof.

We will show that $\pi^{-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F}$
implies
$\pi^{\prime-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F^{\prime}}$.
The other implication can be proved analogously.

Since the $g_{i}$’s and the $h_{j}$’s generate the same ideal $J$ of
$\mathscr{O}_{\mathbbm{C}^{n},0}$, we can write for all $i$,
$g_{i}=\sum_{j}r_{ji}h_{j}$ with some
$r_{ji}\in\mathscr{O}_{\mathbbm{C}^{n},0}$ and at least one
$r_{ji}\not\in{\mathfrak{m}}$. We construct an homomorphism
$\Phi:S^{\prime}\rightarrow S$ of graded
$\mathscr{O}_{\mathbbm{C}^{n},0}$-algebras (preserving degrees) induced by
$\beta_{j}\mapsto\sum_{i}r_{ji}\alpha_{i}$. This gives a natural morphism of
$\mathscr{O}_{\mathbbm{C}^{n},0}$-schemes,
$\varphi:U\rightarrow{\mathbbm{P}}^{u-1}$, where $U\subset{\mathbbm{P}}^{q-1}$
is the complement of the indeterminacy locus of $\varphi$, given by
$V(\Phi(\beta_{1}),\ldots,\Phi(\beta_{u}))\subset{\mathbbm{P}}^{q-1}$. Observe
that $\pi^{-1}({\mathfrak{m}})$ is not contained in the indeterminacy locus of
$\varphi$ because $\Phi(\beta_{j})\not\in{\mathfrak{m}}S$ for at least one
$j$. Notice also that the construction of $\varphi$ imply both
$\pi=\pi^{\prime}\circ\varphi$ and $\sigma|_{U}=\varphi^{*}(\sigma^{\prime})$.
In particular, $\pi^{*}(T(J))=\varphi^{*}\pi^{\prime*}(T(J))$ and
$\mathcal{T}(\sigma)=\varphi^{*}(\mathcal{T}(\sigma^{\prime}))$ on $U$. We
deduce $\mathscr{F}|_{U}=\varphi^{*}\mathscr{F}^{\prime}$.

Assume $\pi^{-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F}$. As
observed above there exists some $P\in\pi^{-1}({\mathfrak{m}})\cap U$,
$P\not\in\mbox{Supp}\mathscr{F}$. Since
$U\cap\mbox{Supp}\mathscr{F}=\varphi^{-1}(\mbox{Supp}\mathscr{F}^{\prime})$,
we have $\varphi(P)\not\in\mbox{Supp}\mathscr{F^{\prime}}$ and
${\mathfrak{m}}S^{\prime}\subset\Phi^{-1}({\mathfrak{m}}S)\subset\Phi^{-1}(P)=\varphi(P)$.
Therefore $\varphi(P)\in\pi^{\prime-1}({\mathfrak{m}})$, showing that
$\pi^{\prime-1}({\mathfrak{m}})\not\subset\mbox{Supp}\mathscr{F^{\prime}}$.∎

Now that we have introduced the notions of $T$-fullness and $T$-dependence, we
present some examples showing that these two notions are not related.

######  Example 3.9.

Let
$J={\mathfrak{m}}^{3}=(x^{3},x^{2}y,xy^{2},y^{3})\subset\mathbbm{C}\\{x,y\\}=\mathscr{O}_{\mathbbm{C}^{2},0}$.
Clearly $J$ is $T$-full. Let us show that $J$ fails to be $T$-dependent.

Notations being as before, $\pi^{*}(T(J))$ is the sheaf associated to
$T(J)[\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}]=(x^{2},xy,y^{2})[\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}].$
Let
$\sigma=x^{3}\alpha_{1}+x^{2}y\alpha_{2}+xy^{2}\alpha_{3}+y^{3}\alpha_{4}$.
Then

$x^{3}\alpha_{1}+x^{2}y\alpha_{2}+xy^{2}\alpha_{3}+y^{3}\alpha_{4},\,3x^{2}\alpha_{1}+2xy\alpha_{2}+y^{2}\alpha_{3},\,x^{2}\alpha_{2}+2xy\alpha_{3}+3y^{2}\alpha_{4}$

are generators of $\mathscr{T}(\sigma)$. In order to investigate the support
of $\mathscr{F}=\pi^{*}(T(J))/\mathscr{T}(\sigma)$ in ${\mathbbm{P}}^{3}$ we
list below the generators of $(\mathscr{T}(\sigma):\pi^{*}(T(J)))$, obtained
with help of _SINGULAR_ , [DGPS].

$3x^{2}\alpha_{1}+2xy\alpha_{2}+y^{2}\alpha_{3}$,

$x^{2}\alpha_{2}+2xy\alpha_{3}+3y^{2}\alpha_{4}$,

$2x\alpha_{2}^{2}-6x\alpha_{1}\alpha_{3}+y\alpha_{2}\alpha_{3}-9y\alpha_{1}\alpha_{4}$,

$x\alpha_{2}\alpha_{3}+2y\alpha_{3}^{2}-9x\alpha_{1}\alpha_{4}-6y\alpha_{2}\alpha_{4}$,

$2x\alpha_{1}\alpha_{3}^{2}+y\alpha_{2}\alpha_{3}^{2}-6x\alpha_{1}\alpha_{2}\alpha_{4}-4y\alpha_{2}^{2}\alpha_{4}+3y\alpha_{1}\alpha_{3}\alpha_{4}$,

$y\alpha_{2}^{2}\alpha_{3}^{2}-4y\alpha_{1}\alpha_{3}^{3}-6x\alpha_{1}\alpha_{2}^{2}\alpha_{4}-4y\alpha_{2}^{3}\alpha_{4}+18x\alpha_{1}^{2}\alpha_{3}\alpha_{4}+15y\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}.$

Clearly $\pi^{-1}({\mathfrak{m}})\subseteq\mbox{Supp}\mathscr{F}$. Hence $J$
is not $T$-dependent.

######  Example 3.10.

Let
$J=(x^{4}+y^{3},xy)\subset\mathbbm{C}\\{x,y\\}=\mathscr{O}_{\mathbbm{C}^{2},0}$.
We have seen before (cf. Example 3.2) that $J$ fails to be $T$-full. Let us
show that $J$ is $T$-dependent.

As above, $\pi^{*}(T(J))$ is the sheaf associated to
$T(J)[\alpha_{1},\alpha_{2}]={\mathfrak{m}}[\alpha_{1},\alpha_{2}].$ Let
$\sigma=(x^{4}+y^{3})\alpha_{1}+xy\alpha_{2}$. Then

$(x^{4}+y^{3})\alpha_{1}+xy\alpha_{2},\,4x^{3}\alpha_{1}+y\alpha_{2},\,3y^{2}\alpha_{1}+x\alpha_{2}$

are generators of $\mathscr{T}(\sigma)$. We investigate the support of
$\mathscr{F}=\pi^{*}(T(J))/\mathscr{T}(\sigma)$ in ${\mathbbm{P}}^{1}$. Below
we present the generators of $(\mathscr{T}(\sigma):\pi^{*}(T(J)))$, obtained
with help of _SINGULAR_ , [DGPS].

$x\alpha_{2}+3y^{2}\alpha_{1}$,

$y\alpha_{2}+4x^{3}\alpha_{1}$,

$\alpha_{2}^{2}-12x^{2}y\alpha_{1}^{2}$,

$(3y^{3}-4x^{4})\alpha_{1}$,

$x^{4}\alpha_{1}$,

$x^{3}y^{2}\alpha_{1}$,

$x^{3}y\alpha_{1}^{2}$,

$x^{2}y^{2}\alpha_{1}^{2}.$

In this case we see that
$(\mathscr{T}(\sigma):\pi^{*}T(J))\not\subseteq{\mathfrak{m}}S$, showing that
$J$ is $T$-dependent.

######  Example 3.11.

If $J\subseteq\mathscr{O}_{\mathbbm{C}^{n},0}$ is a principal ideal then $J$
is $T$-dependent. Indeed, $\mbox{Supp}\mathscr{F}$ is empty, opposite to
$\pi^{-1}({\mathfrak{m}})$ which is not.

###  3.3. Main Result

We are ready to state and prove our result characterizing Tjurina ideals.

######  Theorem 3.12.

Let $I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ be an ideal. Then $I$ is a
Tjurina ideal if and only if $I$ is $T$-full and $\Delta(I)$ is $T$-dependent.

###### Proof.

To begin with, for any
$(\lambda_{1},\ldots,\lambda_{q})\in\mathbbm{C}^{q}\setminus\\{0\\}$ we will
denote by $p_{\lambda}\subseteq\mathbbm{C}[\alpha_{1},\ldots,\alpha_{q}]$ the
homogeneous maximal ideal generated by
$(\\{\lambda_{k}\alpha_{l}-\lambda_{l}\alpha_{k}\\}_{k,l})$. Let
$\Delta(I)=(g_{1},\ldots,g_{q})$ and let $\mathscr{F}$ denote the sheaf
$\pi^{*}(T(\Delta(I)))/\mathscr{T}(\sigma)$ on ${\mathbbm{P}}^{q-1}$, as
described before. For any $P\in D_{+}(\alpha_{l})\subset{\mathbbm{P}}^{q-1}$
one checks easily that
$\mathscr{T}(\sigma)_{P}=T\Big{(}\sigma/\alpha_{l}\Big{)}S_{(P)}.$ Moreover if
$P\supseteq p_{\lambda}S$, then we have the Estimate below. Parenthetical
notation $-_{(P)}$ indicates the submodule of degree zero elements of the
localization at $P$ of a graded module.

(3.1)
$T\Big{(}\sum_{k}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}-\frac{\alpha_{k}}{\alpha_{l}}\Big{)}g_{k}\Big{)}S_{(P)}\subseteq\\\
\subseteq\sum_{k}T\Big{(}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}-\frac{\alpha_{k}}{\alpha_{l}}\Big{)}g_{k}\Big{)}S_{(P)}\subseteq\sum_{k}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}-\frac{\alpha_{k}}{\alpha_{l}}\Big{)}T(g_{k})S_{(P)}\subseteq
IP_{(P)}.$

To prove the direct affirmation in the statement, assume that $I$ is a Tjurina
ideal. We have already seen that $I$ is $T$-full (cf. Proposition 3.5); in
particular $\mathscr{F}=\pi^{*}(I)/\mathscr{T}(\sigma)$. Now we show that
$\Delta(I)$ is $T$-dependent. This is clear if $I$ is $(0)$ because then
$\mbox{Supp}\mathscr{F}$ is empty; so, we assume otherwise. Let $I=T(f)$ with
$f=\sum_{k}\lambda_{k}g_{k}$ for certain
$\lambda_{k}\in\mathscr{O}_{\mathbbm{C}^{n},0}$. Because of Remark 2.5,(v) we
check that $f\not\in{\mathfrak{m}}\Delta(I)$. Hence, at least one of the
$\lambda_{k}$ is an invertible element of $\mathscr{O}_{\mathbbm{C}^{n},0}$.
Using Remark 2.1, we can (and do) assume
$(\lambda_{1},\ldots,\lambda_{q})\in\mathbbm{C}^{q}\setminus\\{0\\}$. Consider
$P=p_{\lambda}S+{\mathfrak{m}}S\in{\mathbbm{P}}^{q-1}$. Then
$P\in\pi^{-1}({\mathfrak{m}})$, so that $\pi^{*}(I)_{P}=I_{\pi(P)}\otimes
S_{(P)}=I_{{\mathfrak{m}}}\otimes S_{(P)}=IS_{(P)}$. On the other hand (say if
$P\in D_{+}(\alpha_{l})$), the assumption $I=T(f)$ together with the Estimate
(3.1) above produces

$IS_{(P)}=T\Big{(}\sum_{k}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}\Big{)}g_{k}\Big{)}S_{(P)}\subseteq\\\
\subseteq
T\Big{(}\sum_{k}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}-\frac{\alpha_{k}}{\alpha_{l}}\Big{)}g_{k}\Big{)}S_{(P)}+T\Big{(}\sigma/\alpha_{l}\Big{)}S_{(P)}\subseteq
IP_{(P)}+\mathscr{T}(\sigma)_{P}\subseteq IS_{(P)}.$

It follows that $IP_{(P)}+\mathscr{T}(\sigma)_{P}=IS_{(P)}$. By Nakayama’s
Lemma we obtain $\pi^{*}(I)_{P}=\mathscr{T}(\sigma)_{P}$. Hence
$P\not\in\mbox{Supp}\mathscr{F}$, which completes the proof that $\Delta(I)$
is $T$-dependent.

To prove the reverse affirmation in the statement $I$ is assumed to be
$T$-full and $\Delta(I)$ is assumed to be $T$-dependent. Then $I=T(\Delta(I))$
and there is some
$P\in\pi^{-1}({\mathfrak{m}})\setminus\mbox{Supp}\mathscr{F}$. It follows that
$\pi^{*}(I)_{P}=\pi^{*}(T(\Delta(I)))_{P}=\mathscr{T}(\sigma)_{P}$. Notice
that out of the decomposition
$S=\mathbbm{C}[\alpha_{1},\ldots,\alpha_{q}]+{\mathfrak{m}}S$ we obtain
$P=P\cap\mathbbm{C}[\alpha_{1},\ldots,\alpha_{q}]+P\cap{\mathfrak{m}}S=P\cap\mathbbm{C}[\alpha_{1},\ldots,\alpha_{q}]+{\mathfrak{m}}S$.
Hence, for any
$(\lambda_{1},\ldots,\lambda_{q})\in\mathbbm{C}^{q}\setminus\\{0\\}$
satisfying $p_{\lambda}\supseteq
P\cap\mathbbm{C}[\alpha_{1},\ldots,\alpha_{q}]$ the homogeneous prime ideal
$p_{\lambda}S+{\mathfrak{m}}S$ also belongs to
$\pi^{-1}({\mathfrak{m}})\setminus\mbox{Supp}\mathscr{F}$.

With this simplification we have - assuming $P=p_{\lambda}S+{\mathfrak{m}}S\in
D_{+}(\alpha_{l})$, as before - $IS_{(P)}=T(\sigma/\alpha_{l})S_{(P)}.$
Estimate (3.1) now gives

$IS_{(P)}\subseteq
T\Big{(}\sum_{k}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}-\frac{\alpha_{k}}{\alpha_{l}}\Big{)}g_{k}\Big{)}S_{(P)}+T\Big{(}\sum_{k}\Big{(}\frac{\lambda_{k}}{\lambda_{l}}\Big{)}g_{k}\Big{)}S_{(P)}\subseteq\\\
\subseteq IP_{(P)}+T(\sum_{k}\lambda_{k}g_{k})S_{(P)}\subseteq IS_{(P)}.$

We obtain $IP_{(P)}+T(\sum_{k}\lambda_{k}g_{k})S_{(P)}=IS_{(P)}$ and by
Nakayama’s Lemma again we deduce
$T(\sum_{k}\lambda_{k}g_{k})S_{(P)}=IS_{(P)}$.

It remains to verify that $I\subseteq T(\sum_{k}\lambda_{k}g_{k})$. Clearly it
suffices to check that if $h\in I$ and there are some $u\in
T(\sum_{k}\lambda_{k}g_{k})$ and $G_{1},G_{2}\in S$, homogeneous of same
degree with $G_{2}\not\in P$ such that $G_{2}h=G_{1}u$, then $h\in
T(\sum_{k}\lambda_{k}g_{k})$. With this reduction, a simple argument involving
divisibility and degree in the UFD $S$ then shows $zh=uH$, for some
homogeneous element $H\in S$ and some degree zero factor $z$ of $G_{2}$. In
particular, $z\in\mathscr{O}_{\mathbbm{C}^{n},0}\setminus{\mathfrak{m}}$.
Again by degree reasons, $H$ has degree zero, ie, it belongs to
$\mathscr{O}_{\mathbbm{C}^{n},0}$. This shows that $h\in
T(\sum_{k}\lambda_{k}g_{k})$. Hence $I\subseteq T(\sum_{k}\lambda_{k}g_{k})$
and we are done.∎

######  Corollary 3.13.

Suppose $I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ is a non-zero Tjurina ideal
and let $\Delta(I)=(g_{1},\ldots,g_{q})$. Then $I=T(\sum_{k}\lambda_{k}g_{k})$
for $(\lambda_{1}:\ldots:\lambda_{q})$ in a non empty Zariski open subset of
${\mathbbm{P}}_{\mathbbm{C}}^{q-1}$.

###### Proof.

The canonical inclusion
$\mathbbm{C}\rightarrow\mathscr{O}_{\mathbbm{C}^{n},0}$ induces
$\mathbbm{C}[\alpha_{1},\ldots,\alpha_{q}]\rightarrow S$, a graded inclusion
of $\mathbbm{C}$-algebras; hence a morphism
$\rho:{\mathbbm{P}}^{q-1}\rightarrow{\mathbbm{P}}_{\mathbbm{C}}^{q-1}$, which
is proper and dominant. Notation being as in the proof of Theorem 3.12, if $I$
is a Tjurina ideal then
$\pi^{-1}({\mathfrak{m}})\setminus\mbox{Supp}\mathscr{F}$ is a non empty
Zariski open subset of $\pi^{-1}({\mathfrak{m}})$. It follows that its image
under $\rho$ is a non-empty open Zariski subset of
$U\subset{\mathbbm{P}}_{\mathbbm{C}}^{q-1}$. Closed points
$(\lambda_{1}:\ldots:\lambda_{q})$ of $U$ correspond to prime ideals of type
$p_{\lambda}S+{\mathfrak{m}}S\in\pi^{-1}({\mathfrak{m}})\setminus\mbox{Supp}\mathscr{F}$.
Then proceed as in the proof of the previous Theorem.∎

##  4\. Remarks and Examples

Now that we achieved a complete characterization of Tjurina ideals by means of
Theorem 3.12, we include some observations regarding intersections of Tjurina
ideals. We focus on monomial ideals, since it already contains the kind of
phenomenon we aim to illustrate.

######  Example 4.1.

_Irreducible_ monomial ideals are necessarily Tjurina ideals. Recall that
irreducible monomial ideals are of the form
$(x_{i_{1}}^{e_{1}},\ldots,x_{i_{s}}^{e_{s}})$, for
$\\{i_{1}<\ldots<i_{s}\\}\subseteq\\{1,\ldots,n\\}$ and all $e_{j}>0$. Hence

$(x_{i_{1}}^{e_{1}},\ldots,x_{i_{s}}^{e_{s}})=T(x_{i_{1}}^{e_{1}+1}+\ldots+x_{i_{s}}^{e_{s}+1}).$

The following well-known result (see [HH], Theorem 1.3.1, Corollary 1.3.2, for
instance) motivates us to ask questions about monomial ideals which are also
Tjurina ideals.

######  Proposition 4.2.

Any monomial ideal $I\subset\mathscr{O}_{\mathbbm{C}^{n},0}$ can be written
uniquely as a finite and irredundant intersection $I=\cap_{k=1}^{m}\,Q_{k}$ of
_irreducible_ monomial ideals.

Of course we regard the irreducible monomial ideals as the building blocks of
more general monomial ideals. Unfortunately, the result of the previous
Example does not extend to arbitrary monomial ideals:
$(x^{2},xy,y^{2})=(x^{2},y)\cap(x,y^{2})\subset\mathscr{O}_{\mathbbm{C}^{2},0}=\mathbbm{C}\\{x,y\\}$
is not a Tjurina ideal (cf. Example 1.4); even for principal monomial ideals
such a result does not hold (cf. Proposition 3.3).

This is _not_ to say that reducible monomial ideals are never Tjurina ideals.
For instance,
$I=(yz,xz,xy)=(x,y)\cap(x,z)\cap(y,z)\subset\mathscr{O}_{\mathbbm{C}^{3},0}=\mathbbm{C}\\{x,y,z\\}$
of Example 1.5 is a Tjurina ideal.

In our computations of Example 4.5, the next example will be useful.

######  Example 4.3.

Let $I=(x_{i_{1}}^{e_{1}},\ldots,x_{i_{s}}^{e_{s}})$ be an irreducible
monomial ideal as in Example 4.1. From the results in Section 2, discussion
before Proposition 2.11 we deduce that
$a_{1}x_{i_{1}}^{e_{1}}+\ldots+a_{s}x_{i_{s}}^{e_{s}}\in\Delta(I)$ if and only
if $a_{k}x_{i_{k}}^{e_{k}-1}\in I$, for all $k=1,\ldots,s$. Here we used that
$I$ is monomial. This, in turn, happens if and only if
$a_{k}\in(I:x_{i_{k}}^{e_{k}-1})=(x_{i_{1}}^{e_{1}},\ldots,x_{i_{k}},\ldots,x_{i_{s}}^{e_{s}})$.
As a result we have
$\Delta(I)=(x_{i_{1}}^{e_{1}+1},\ldots,x_{i_{s}}^{e_{s}+1})+I^{2}$, a monomial
ideal.

######  Remark 4.4.

The previous Example together with Proposition 4.2 and Remark 2.5, (iii) shows
that the ideal of antiderivatives of a monomial ideal is also monomial.

It seems worth examining the Example below, in which a monomial and reducible
Tjurina ideal is presented, together with some considerations regarding the
dynamics of formation of Tjurina ideals. The Example also provides an example
of a Tjurina ideal with an embedded component of codimension $2$, so answering
negatively a Question left by Aluffi ([A], Example 3.1).

######  Example 4.5.

Let
$f=xw^{2}-yz^{2}\in\mathscr{O}_{\mathbbm{C}^{4},0}=\mathbbm{C}\\{x,y,z,w\\}$.
It has a monomial Tjurina ideal given by $T(f)=(yz,z^{2},xw,w^{2})=I_{1}\cap
I_{2}\cap I_{3}\cap I_{4}$, where $I_{1}=(z,w)$, $I_{2}=(y,z^{2},w)$,
$I_{3}=(x,z,w^{2})$ and $I_{4}=(x,y,z^{2},w^{2})$. The ideal $T(f)$ has an
embedded component of codimension $2$ at $(x,y,z^{2},w^{2})$.

We notice that $I_{1}\cap I_{2}\cap I_{3}\cap I_{4}$ is a Tjurina ideal but
that some partial intersections are not. For instance $I_{1}\cap I_{4}$ or
$I_{2}\cap I_{3}$ are not Tjurina ideals as one can easily verify computing
numbers of generators: both are minimally $6$-generated.

Harder is to check that $I_{1}\cap I_{2}\cap I_{3}=(yz,z^{2},xw,zw,w^{2})$ is
not a Tjurina ideal. We begin computing its ideal of antiderivatives
$\Delta(I_{1}\cap I_{2}\cap
I_{3})=\Delta(I_{1})\cap\Delta(I_{2})\cap\Delta(I_{3})$. Using the result of
the previous Example we get $\Delta(I_{1})=(z^{2},w^{2},zw),\,\,\,$
$\Delta(I_{2})=(y^{2},z^{3},w^{2},yz^{2},yw,z^{2}w),\,\,\,$ and
$\Delta(I_{3})=(x^{2},z^{2},w^{3},xz,xw^{2},zw^{2}).$ Hence

$\Delta(I_{1}\cap I_{2}\cap
I_{3})=(yz^{2},w^{3},z^{2}w,xw^{2},zw^{2},w^{3},xyzw).$

It is clear now that $I_{1}\cap I_{2}\cap I_{3}$ is $T$-full but we can check
with SINGULAR [DGPS] that $\Delta(I_{1}\cap I_{2}\cap I_{3})$ is not
$T$-dependent. Then Theorem 3.12, guarantees that $I_{1}\cap I_{2}\cap I_{3}$
is not a Tjurina ideal.

Acknowledgements: The author is grateful to Marcelo Escudeiro Hernandes, whose
interest led to the investigation in the present work.

## References

  * [A] Aluffi, P. - Singular schemes of hypersurfaces. Duke Mathematical Journal 80.2 (1995): 325-351.
  * [AM] Atiyah, M. F., Macdonald, I. G. - Introduction to Commutative Algebra. (Addison-Wesley Series in Mathematics). Westview Press Incorporated, (1969).
  * [B] Boubakri, Y. - Hypersurface Singularities in Positive Characteristic. PhD. Thesis, Technischen Universität Kaiserslautern (2009).
  * [DGPS] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H. -  Singular 4-3-0 — A computer algebra system for polynomial computations.  http://www.singular.uni-kl.de (2022).
  * [GLS] Greuel, G. M., Lossen, C., Shustin, E. I.-Introduction to singularities and deformations. Springer Science Business Media(2007).
  * [H1] Hartshorne, R. Deformation theory. Vol. 257. New York: Springer, (2010).
  * [H2] Hartshorne, R. Algebraic geometry. Graduate texts in mathematics 52 (1977).
  * [HH] Herzog, J., Hibi, T.. Monomial ideals. Graduate texts in mathematics 260 (2010).
  * [MY] Mather, John N., and Stephen S-T. Yau. -Classification of isolated hypersurface singularities by their moduli algebras. Inventiones mathematicae 69.2: 243-251 (1982).

Author address:

Instituto de Matemática e Estatística
Universidade Federal Fluminense
R. Prof. Marcos Waldemar de Freitas Reis, Bloco G
Campus do Gragoatá - São Domingos
24.210-201 - Niterói, RJ
Brazil

e-mail<EMAIL_ADDRESS>


InfoBox

# Footprint of Publication Selection Bias on Meta-Analyses in Medicine,
Environmental Sciences, Psychology, and Economics

František Bartoš1,2*, Maximilian Maier3*, Eric-Jan Wagenmakers1,
Franziska Nippold1, Hristos Doucouliagos4, John P. A. Ioannidis5,6,7,8,9,
Willem M. Otte10, Martina Sladekova11, Teshome K. Deresssa12, Stephan B.
Bruns12,13
Daniele Fanelli14,15, T.D. Stanley4

∗Both authors contributed equally
1 Department of Psychological Methods, University of Amsterdam
2 Institute of Computer Science, Czech Academy of Sciences
3 Department of Experimental Psychology, University College London
4 Department of Economics, Deakin University
5 Meta-Research Innovation Center at Stanford (METRICS)
6 Department of Epidemiology and Population Health, Stanford University School
of Medicine
7 Stanford Prevention Research Center, Department of Medicine, Stanford
University School of Medicine
8 Department of Biomedical Data Science, Stanford University School of
Medicine
9 Department of Statistics, Stanford University School of Humanities and
Sciences
10 Department of Pediatric Neurology, UMC Utrecht Brain Center,
University Medical Center Utrecht, and Utrecht University
11 School of Psychology, University of Sussex
12 Centre for Environmental Sciences, Hasselt University
13 Department of Economics, University of Göttingen
14 Department of Methodology, London School of Economics and Political Science
15 School of Social Sciences, Heriot-Watt University

Correspondence concerning this article should be addressed to František Bartoš
at<EMAIL_ADDRESS>

###### Abstract

Publication selection bias undermines the systematic accumulation of evidence.
To assess the extent of this problem, we survey over 68,000 meta-analyses
containing over 700,000 effect size estimates from medicine (67,386/597,699),
environmental sciences (199/12,707), psychology (605/23,563), and economics
(327/91,421). Our results indicate that meta-analyses in economics are the
most severely contaminated by publication selection bias, closely followed by
meta-analyses in environmental sciences and psychology, whereas meta-analyses
in medicine are contaminated the least. After adjusting for publication
selection bias, the median probability of the presence of an effect decreased
from 99.9% to 29.7% in economics, from 98.9% to 55.7% in psychology, from
99.8% to 70.7% in environmental sciences, and from 38.0% to 29.7% in medicine.
The median absolute effect sizes (in terms of standardized mean differences)
decreased from $d=0.20$ to $d=0.07$ in economics, from $d=0.37$ to $d=0.26$ in
psychology, from $d=0.62$ to $d=0.43$ in environmental sciences, and from
$d=0.24$ to $d=0.13$ in medicine.

##  1 Introduction

Publication selection biases (PSB) are defined as the selective reporting of
results in ways that deviate from the objective, complete scientific record.
PSB may entail the suppression of “negative” findings or the conversion of
“negative” results into more “positive” ones (e.g., those with more favorable
$p$-values and/or with larger effect sizes) and might represent a problem in
all scientific disciplines, e.g., [1, 2, 3, 4, 5]. Studies that examine the
self-reported behavior of researchers show that 78% of researchers failed to
report all dependent measures of a study [6] (however, see [7], for a response
that suggests a lower proportion). Some studies also suggest that PSB might be
modestly increasing in some areas, although the exact nature, prevalence, and
impact of PSB is unknown and likely to be variable across scientific fields
[8, 9].

To gauge the extent of the PSB, one would need to have access to the complete
scientific record or a representative and wide-coverage sample of it. However,
this is infeasible as much of the relevant data is not publicly recorded.
Instead, the footprint of PSB is indirectly probed by re-analyzing meta-
analyses in several specific fields with different statistical techniques [9,
10, 11, 12, 13, 14] and focusing on patterns in the published results that
would herald the presence of PSB. All these available methods try to identify
the footprint of PSB, and thus their results need to be interpreted with
caution since these patterns (e.g., correlations of effect sizes and standard
errors) may sometimes be due to factors other than PSB (e.g., genuine
heterogeneity across studies). However, when large numbers of meta-analyses
show the same patterns, this constitutes a probable footprint of PSB, which
can be used to estimate its relative magnitude across different fields.

Previous field-wide assessments of PSB suggested that the prevalence of over-
reporting positive results and other possible symptoms of bias increased
moving from the physical to the biological and the social sciences, and even
suggested that problems might be worsening over time in the latter [9, 15, 16,
17, 18, 19, 20]. However, these estimates were based on proxy measures of PSB
that have several limitations.

To our knowledge, no previous survey of the potential footprint of PSB has
used state-of-the-art methods. Our proposed approach is more comprehensive
than past surveys: employing different strategies to identify potential PSB,
using new measures of PSB, and analyzing a much larger number of research
studies covering the fields of medicine, environmental sciences, psychology,
and economics.

##  2 Methods

###  2.1 Data Sets

We used five large data sets from medicine, environmental sciences,
psychology, and economics. The data set from medicine comprises meta-analyses
of continuous and dichotomous outcomes obtained from the Cochrane Database of
Systematic Reviews published between 1997 and 2020. The data set from
environmental sciences comprises the meta-analyses of mean differences, odds
ratios, and correlation coefficients by Deressa and colleagues [21] published
between 2010 and 2020. The data sets from psychology comprise the meta-
analyses of mean differences and correlation coefficients by Stanley and
colleagues [12] published between 2011 and 2016 combined with a random sample
of meta-analyses published in psychological journals by Sladekova and
colleagues [22] published in 2008 and 2018. Finally, the data set from
economics comprises the extended data set of meta-analyses of regression and
correlation coefficients by Ioannidis and colleagues [10] published between
1967 and 2021. Eighty-four meta-analyses were part of both the [10] and [12]
data set. Since each of the meta-analyses could be classified in both fields
(psychology or economics), we did not remove them from either of the data
sets. From each data set we only used meta-analyses with at least three
estimates reported using standardized effect size metrics such as log odds
ratios, standardized mean differences, and (partial) correlation coefficients
that can be transformed to a common standardized mean difference effect size
metric, Cohen’s $d$.

####  2.1.1 Medicine

The data set from medicine comprises meta-analyses of continuous and
dichotomous outcomes obtained from the Cochrane Database of Systematic Reviews
(CSDR) published between 1997 and 2020. We identified systematic reviews in
the CDSR through PubMed, limiting the period to Jan 2000 – May 2020. For that,
we used the NCBI’s EUtils API with the following query: “Cochrane Database
Syst Rev”[journal] AND (“2000/01/01”[PDAT]: “2020/05/31”[PDAT]). For each
review, we downloaded the XML meta-analysis table file (rm5-format) associated
with the review’s latest version. We extracted the tables with continuous and
dichotomous outcomes from these rm5-files with a custom Javascript and R
programs (https://github.com/wmotte/cochrane2022).

We selected meta-analysis tables based on the highest aggregation reported in
the CSDR. For each meta-analysis, we removed estimates based on one or fewer
participants in the control or treatment group and used all meta-analyses with
at least three effect size estimates.

####  2.1.2 Environmental Sciences

The environmental sciences data set consists of meta-analyses of mean
differences, correlation coefficients, and odds ratios published between 2010
and 2020. The literature search was performed in the Scopus database using the
query: "TITLE-ABS-KEY (“meta analy*” OR “meta-analy*” OR “metaanaly*” OR “meta
reg*” OR “meta-reg*” OR “metareg*”) AND SUBJAREA (envi)" on July 21, 2020.
Detailed information about the sampling strategy and inclusion/exclusion
criteria used can be found in [21].

####  2.1.3 Psychology

The data set from psychology comprise the data set of meta-analyses of mean
differences and correlation coefficients of Stanley and colleagues [12]
published between 2011 and 2016 combined with data from Sladekova and
colleagues [22], a random sample of 433 meta-analyses from 90 articles
published in 2008 and 2018. See [12] and [22] for more details about the
collected data sets. None of the meta-analyses by [22] were published in
Psychological Bulletin, precluding overlap with Stanley and colleagues [12]
data set.

####  2.1.4 Economics

The data set from economics comprise the extended data set of meta-analyses of
regression and correlation coefficients of Ioannidis and colleagues [10]
published between 1967 and 2021. The meta-analyses were identified using
various search engines (e.g.. Econlit and Scopus), publisher sites (e.g.,
Science Direct, Sage, and Wiley), webpages of researchers known to publish
meta-analyses, and by searching all volumes of individual journals that are
known to publish meta-analyses. We also emailed 109 research teams (associated
with either sole authored or co-authored meta-analyses) for data, with a 67%
response rate. The search for data ended May 30th, 2021.

We selected meta-analyses of standardized mean differences, (partial)
correlation coefficients, and mean differences (if enough information was
available to compute the standardized mean differences).

####  2.1.5 Effect Size Calculation

In cases where the data set did not already feature standardized effect size
(Cohen’s $d$, correlation coefficient $r$, log(_OR_), or Fisher’s $z$), we
used the metafor R package [23] to calculate the standardized effect sizes.
For dichotomous outcomes with zero cell counts, we used the default empty cell
correction, adding $1/2$ to empty cells. Finally, we converted all
standardized effect sizes to Fisher’s $z$ by using the formulas in [24].

###  2.2 Publication Bias Adjustment  with Bayesian Model-Averaging

We used the PSB detection and correction technique RoBMA-PSMA [25, 26]. RoBMA
employs Bayesian model-averaging [27, 28] and combines the best of two well-
performing publication bias adjustment methods: selection models  with six
different weight functions that adjust for publication selection across a
combination of statistical significance and direction of the effect [29] and
PET-PEESE , which adjusts for the relationship between effect sizes and
standard errors or standard errors squared [30].  Bayesian model-averaging
allows us to combine these publication bias adjustment methods based on their
predictive adequacy, such that models that predict well have a larger impact
on the inference. In that way, we can evaluate the evidence in favor or
against the hypothesis of PSB and its impact without committing to any single
estimation or correction method [27].

We used the default RoBMA parameterization which was shown to achieve better
performance in both simulation studies and real data examples than either of
publication bias adjustment methods alone [26]. It gives equal prior model
probabilities to models assuming the presence vs. absence of an effect,
heterogeneity, and publication selection bias. RoBMA employs a standard normal
distribution on the effect size, $\mu\sim\text{Normal}(0,1)$, empirically
informed Inverse-gamma distribution on the heterogeneity,
$\tau\sim\text{Inverse-Gamma}(1,0.15)$ [31], cumulative unit Dirichlet prior
distributions on publication probabilities, and Cauchy prior distributions on
the PET-PEESE regression coefficients, $\text{PET}\sim\text{Cauchy}_{+}(0,1)$,
$\text{PEESE}\sim\text{Cauchy}_{+}(0,5)$.

[H] Bayes factors The Bayes factor is the key inference criterion for much of
Bayesian statistics, e.g., [32, 33]. It compares the relative predictive
accuracy (i.e., likelihood of the data) under competing hypotheses (e.g.,
$\mathcal{H}_{1}$ vs. $\mathcal{H}_{0}$) and it can also be expressed as the
ratio of prior and posterior model odds,
$\underbrace{\frac{p(\text{data}\mid\mathcal{H}_{1})}{p(\text{data}\mid\mathcal{H}_{0})}}_{\begin{subarray}{c}\text{Bayes
factor}\end{subarray}}=\underbrace{\frac{p(\mathcal{H}_{1}\mid\text{data})}{p(\mathcal{H}_{0}\mid\text{data})}}_{\begin{subarray}{c}\text{Posterior
odds}\end{subarray}}\,\,\bigg{/}\underbrace{\frac{p(\mathcal{H}_{1})}{p(\mathcal{H}_{0})}}_{\begin{subarray}{c}\text{Prior
odds}\end{subarray}}.$ Although the Bayes factor is a continuous measure of
strength of evidence, the following rules of thumb may aid interpretation:
Bayes factors between 1 and 3 are commonly regarded as weak evidence, Bayes
factors between 3 and 10 as moderate evidence, and Bayes factors larger 10 as
strong evidence for the alternative (or the hypothesis at the top of Equation
2.2). When the evidence for the null is considered, the Bayes factor is simply
inverted. In other words, a Bayes factor between 1/3 and 1 is considered weak
evidence, a Bayes factor between 1/10 and 1/3 moderate and smaller 1/10 strong
evidence for the null (e.g., [34]; [35]).

###  2.3 Measures

For each meta-analysis, we used RoBMA to calculate the (PSB) adjusted
posterior model-averaged effect size assuming it is present (i.e., without
averaging over the point null models to reduce shrinkage toward zero),
$\mu_{\text{adj},k}$; publication bias adjusted posterior probability of the
presence of the effect,
$p_{\text{adj},k}(\mathcal{H}_{1}\mid\text{data}_{k})$; and the posterior
probability of the presence of PSB,
$p_{\text{adj},k}(\mathcal{H}_{\text{psb}}\mid\text{data}_{k})$. To isolate
the effect of PSB adjustment, we compare the Bayesian, PSB unadjusted, model-
averaged meta-analysis by dropping the PSB adjustment and thereby estimating
the unadjusted posterior probability of the presence of the effect assuming it
is present, $p_{\text{unadj},k}(\mathcal{H}_{1}\mid\text{data}_{k})$.
$k=1,\dots,K$ to denotes the individual meta-analyses. Each meta-analysis is
based on $N_{k}$ estimates that are characterized with data describing the
effect size $y_{k,n}$ and standard error $\text{se}_{k,n}$.

####  2.3.1 Evidence for the Effect

We used the change in the posterior probability of the effect and the
(standardized) evidence inflation factor to quantify the effect of PSB on
meta-analytic evidence.

The Posterior probability of the effect is an intuitive way of quantifying the
evidence in favor of the alternative hypothesis of the presence of an effect.
Under the assumption of equal prior probability of the presence and the
absence of the effect, $p(\mathcal{H}_{1})=p(\mathcal{H}_{0})$, posterior
probabilities larger than 0.5 indicate that the data are more likely under the
presence of the effect. On the other hand, posterior probabilities lower than
0.5 indicate that the data are more likely under the absence of the effect.
The ability to quantify evidence for both the null and the alternative is a
key benefit of Bayesian methods over null hypothesis significance testing.
[36, 37].

A corresponding way of quantifying the evidence of an effect is via Bayes
factors (see InfoBox 2.2 for more detail). Bayes factors quantify the change
from prior to posterior odds for the presence of the effect. The advantage of
Bayes factors is that they are independent of the prior odds for the presence
of the effect. In other words, Bayes factors isolate the evidence for the
presence of the effect contained in the data. In our settings, the assumption
of equal prior probabilities leads to an equivalence between Bayes factors and
posterior odds.

The change from the PSB unadjusted posterior probability of the effect,
$p_{\text{unadj},k}(\mathcal{H}_{1}\mid\text{data}_{k})$, to the PSB adjusted
posterior probability of the effect,
$p_{\text{adj},k}(\mathcal{H}_{1}\mid\text{data}_{k})$, quantifies the amount
of evidence introduced by PSB. The larger the impact of PSB, the larger the
difference between the PSB unadjusted and PSB adjusted posterior probabilities
of the effect. If there was no PSB, we would observe no change in the
posterior probability of the effect after PSB adjustment.

Evidence inflation factor (EIF) quantifies the degree to which the evidence in
favor of the presence of the effect was inflated due to PSB. $\text{EIF}_{k}$
quantifies the amount of evidence in favor of the effect in the PSB unadjusted
meta-analysis, $\text{BF}_{10,\text{unadj},k}$, to the amount of evidence in
favor of the effect in the PSB adjusted meta-analysis
$\text{BF}_{10,\text{adj},k}$,

$\text{EIF}_{k}=\frac{\text{BF}_{10,\text{unadj},k}}{\text{BF}_{10,\text{adj},k}}.$
(1)

An evidence inflation factor larger than one indicates inflated evidence in
favor of the effect due to PSB.

However, the amount of evidence contained in each meta-analysis, and the
corresponding evidence inflation, is dependent on the number of meta-analyzed
estimates, $N_{n}$, i.e., more estimates leads to more evidence. To facilitate
the comparison of evidence inflation due to PSB in meta-analyses with
different numbers of estimates, we also compute the standardized, per-
estimate, evidence inflation factor in each meta-analysis, $\text{sEIF}_{k}$,
by standardizing the EIF by the number of estimates,

$\text{sEIF}_{k}=\text{EIF}_{k}^{\frac{1}{N_{k}}},$ (2)

where sEIF represents each estimate’s marginal contribution, on average, to
the evidence inflation due to PSB. The sEIF also partially mitigates the
potential issue of dependent estimates within a meta-analysis. In the most
extreme case, e.g., identical estimates, the same data is conditioned upon
multiple times, which leads to overestimation of evidence. Taking only a
fraction of each estimate’s likelihood, proportional to the number of
estimates, then ensures that the data are not conditioned upon more than once,
although data producing multiple estimates are still weighted more heavily.

####  2.3.2 Effect Size Estimates

Absolute bias (bias) quantifies the degree to which the average effect sizes
in each meta-analysis,

$\hat{y}_{k}=\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}y_{k,n},$

overestimates the PSB-adjusted meta-analytic effect size estimate assuming the
presence of the effect $\mu_{\text{adj},k}$,

$\text{bias}_{k}=\hat{y}_{k}-\mu_{\text{adj},k}.$ (3)

Absolute bias larger than zero indicates that PSB leads to inflated effect
size estimates. We compare the average effect sizes to the PSB-adjusted effect
sizes assuming the presence of the effect (conditional effect size estimates)
rather than averaging across all models. Excluding models assuming the absence
of a mean effect mitigates the pooling towards $0$ in meta-analyses more
consistent with the null hypothesis. Tables 5 and 6 in the Supplementary
Materials use the PSB-adjusted effect sizes model-averaged across all models,
including models assuming the absence of a mean effect. These estimates, which
are model-averaged also over the null, indicate stronger absolute bias
compared to the conditional estimates presented in the main manuscript (Table
2).

Overestimation factor (OF) quantifies the degree to which the average effect
sizes in meta-analyses overestimate the PSB-adjusted effect size estimates
assuming the presence of the effect,

$\text{OF}=\frac{\frac{1}{K}\sum_{k=1}^{K}\hat{y}_{k}}{\frac{1}{K}\sum_{k=1}^{K}\mu_{\text{adj},k}}.$
(4)

An overestimation factor larger than one is evidence of PSB. We use delta
method to obtain confidence interval of the overestimation factor. In the
Supplementary Materials, we also report medians and interquartile ranges of
per meta-analysis overestimation factors,

$\text{OF}_{k}=\frac{\hat{y}_{k}}{\mu_{\text{adj},k}}.$ (5)

However, note that $\text{OF}_{k}$ can lead to non-sensible results as a meta-
analysis with a positive mean effect and very small negative PSB-adjusted
effect sizes estimate results in an extremely large negative $\text{OF}_{k}$.

####  2.3.3 Evidence for Publication Selection Bias

Posterior probability of PSB is an intuitive way of quantifying the evidence
in favor of PSB. Similarly to the posterior probability of the presence of the
effect, under the assumption of equal prior probability of the presence and
the absence of PSB,
$p(\mathcal{H}_{\text{PSB}})=p(\mathcal{H}_{\text{NoPSB}})$, a posterior
probability larger than 0.5 indicates that the data are more likely under the
presence of PSB. On the other hand, a posterior probability lower than 0.5
indicates that the data are more likely under the absence of PSB. As before,
the Bayes factor for the presence of PSB, $\text{BF}_{\text{psb}}$,111In other
publications, we abbreviate this as $\text{BF}_{\text{pb}}$ or
$\text{BF}_{\omega\bar{\omega}}.$ quantifies the change from prior to
posterior odds for the presence of PSB. A Bayes factor in favor of the
presence of PSB larger than one provides evidence in favor of the presence of
PSB and lower than one provides evidence against the presence of PSB

Relative publication probabilities quantify the relative probability of an
estimate being published for a given $p$-value interval compared to estimates
with statistically significant $p$-values. We use one-sided $p$-values,
resulting in $p$-values larger than 0.5 corresponding to estimates in the
opposite direction. To facilitate the interpretation we visualize a weight
function that shows the change of relative publication probabilities across
the range of $p$-values. We report the results only in Supplementary
Materials.

Effect size inflation in imprecise estimates quantifies the relationship
between the effect sizes and their standard errors. To facilitate the
interpretation of the funnel asymmetry test, we visualize the bias in effect
sizes as a function of standard errors (incorporating the quadratic term from
the RoBMA model). We report the results only in Supplementary Materials.

##  3 Results

###  3.1 Descriptives

Table 1: Summary of the data sets from each field. The number of estimates per meta-analysis (Estimates/MA) and the unweighted simple mean effect size of estimates within each meta-analysis (Effect Sizes) are reported as medians with the interquartile range (in parentheses). The proportion of the statistically significant (Prop. Significant) meta-analytic effect size estimates is based on a random-effect meta-analysis estimated via restricted maximum likelihood with $\alpha=0.05$ (removing one environmental sciences and 275 medical meta-analyses that did not converge) . Field | Meta-Analyses | Estimates | Estimates/MA | Effect Sizes ($d$) | Prop. Significant
---|---|---|---|---|---
Medicine | 67,386 | 597,699 | 5 (4, 10) | 0.24 (0.09, 0.47) | 0.39
Environmental | 199 | 12,707 | 26 (11, 59) | 0.62 (0.31, 0.95) | 0.85
Psychology | 605 | 23,563 | 18 (9, 40) | 0.37 (0.18, 0.61) | 0.78
Economics | 327 | 91,421 | 66 (30, 283) | 0.20 (0.09, 0.37) | 0.82

Table 1 compares the characteristics of the meta-analyses from each field.
Medical meta-analyses contain the smallest number of estimates per meta-
analysis, followed by psychology and environmental sciences with five to six
times the number of estimates compared to medicine. Finally economics meta-
analyses contain over twelve times the median number of estimates compared to
medicine. Contrary to a naive expectation that more estimates may be conducted
to establish smaller effects, economic and medical effect sizes are
approximately the same magnitude (measured as a mean effect size per meta-
analysis). Effect sizes in psychology are roughly twice as large as those in
economics, and effect sizes in the environmental sciences are approximately
three times larger than those in economics. Differences in the number of
estimates per meta-analysis are most closely reflected in the proportion of
statistically significant random-effects estimates. Notably, random-effects
estimates in economics, psychology, and environmental sciences are
statistically significant approximately twice as often as in medicine. This
disparity in the proportion of statistically significant meta-analyses is
consistent when comparing meta-analyses with a matched number of estimates
across the disciplines, although the difference in mean effects is somewhat
smaller (see Table 1 in the Supplementary Materials).

We summarize results from all meta-analyses, apart from seven medical meta-
analyses that did not converge. See Supplementary Materials for analyses
showing that matching meta-analyses based on the number of primary estimates
within each meta-analysis does not meaningfully affect the conclusions.

###  3.2 Evidence for the Effect

Figure 1: Median, Interquartile Range, and Distribution of Posterior
Probability for the Presence of the Effect Before and After Adjustment for
Publication Selection Bias in Each Field

Note. The width of grey area indicates density, the light grey area indicates
the interquartile range, and the black line indicates the median. The $y$-axis
is scaled according to posterior probabilities assuming equal prior
probabilities of presence vs absence of the effect. See the secondary $y$-axis
for Bayes factors in favor of the effect that are independent of the assumed
prior probability of the effect.

Figure 1 shows medians, interquartile ranges, and distributions of the
posterior probability of an effect before and after adjusting for PSB. These
distributions reveal several patterns. First, meta-analyses in economics,
psychology, and environmental sciences predominantly show evidence for an
effect before adjusting for PSB (unadjusted); whereas meta-analyses in
medicine often display evidence against an effect. This disparity between the
fields remains even when comparing meta-analyses with equal numbers of effect
size estimates (see Supplementary Materials). After correcting for PSB, the
posterior probability of an effect drops much more in economics, psychology,
and environmental sciences (medians drop from $99.9$% to $29.7$%, from $98.9$%
to $55.7$%, and from $99.8$% to $70.7$% , respectively) compared to medicine
($38.0$% to $29.7$%). The pattern is especially striking in economics, where
the median posterior probability of an effect drops by more than seventy
percentage points after PSB correction. Mean decreases in posterior
probabilities show a similar pattern but with somewhat smaller reductions
(Table 9 and 10 in Supplementary Materials).  In all four disciplines,
adjusting for PSB resulted in a substantial decrease in the strength of
evidence for the effect: the proportion of meta-analyses with at least strong
evidence for the presence of an effect (i.e., $\text{BF}_{10}>10$) decreased
from 20.2% to 5.3% in medicine, from 72.4% to 30.7% in environmental sciences,
from 59.8% to 27.3% in psychology, and from 72.8% to 19.6% in economics. A
comparable decrease was also present when comparing the proportion of meta-
analyses with at least moderate evidence for the presence of an effect (i.e.,
$\text{BF}_{10}>3$; from 28.9% to 12.3% in medicine, from 80.4% to 47.3% in
environmental sciences, from 67.4% to 38.39% in psychology, and from 76.8% to
27.6% in economics).

Furthermore, we quantify the inflation of evidence in favor of an effect in
meta-analyses via the evidence inflation factor – the increase in Bayes factor
in favor of the effect due to the PSB. We find that meta-analyses in economics
inflate the evidence by a median factor of 11,369, whereas the meta-analyses
in environmental sciences and psychology inflate the evidence by ‘only’ $45.9$
and $30.0$ respectively, and medicine by a median factor of $1.33$. These
extreme differences between the fields are largely driven by the disparity in
the typical numbers of estimates per meta-analysis across the disciplines
(Table 1). After standardizing the evidence inflation factor (sEIF) by the
number of estimates per meta-analysis, we find that per estimate evidence
inflation is the largest in psychology with a median factor of $1.27$,
followed by environmental sciences with a median factor of $1.22$, economics
with a median factor of $1.15$, and medicine with a median factor of $1.05$.
Again, the strong evidence inflation in economics (11,369) is largely due to
having many more estimates per meta-analysis than psychology and medicine.
However, even after adjusting for different numbers, meta-analyses in medicine
still show the least inflated evidence due to PSB.

###  3.3 Effect Size Estimates

Table 2: Summary of the footprints of publication selection bias on the meta-
analytic effect sizes in the form of absolute bias (in Cohen’s $d$) and
overestimation factor.

Field | Absolute Bias ($d$) | Overestimation Factor
---|---|---
Medicine | 0.13 [0.12, 0.13] | 1.62 [1.60, 1.64]
Environmental | 0.33 [0.24, 0.41] | 1.78 [1.42, 2.13]
Psychology | 0.13 [0.11, 0.14] | 1.39 [1.24, 1.55]
Economics | 0.15 [0.13, 0.17] | 2.16 [1.69, 2.64]

Note. The results are based on the comparison of publication bias adjusted
meta-analytic effect size estimates assuming presence of the effect to the
mean effect sizes per meta-analysis. The table displays means and 95%
confidence intervals. (See Table 4 in the Supplementary Materials for medians
and interquartile ranges.)

Table 2 summarizes the effect of PSB on effect sizes in each field. The first
column reveals that environmental sciences, on average, suffer from as much as
two and a half times larger absolute bias as medicine, economics, or
psychology. The degree of absolute bias in environmental sciences is so large
that it is comparable to average unadjusted effect sizes in other fields.
Otherwise, medicine, psychology, and economics share a comparable degree of
absolute bias. The median absolute bias in each field is lower than the mean
bias due to the right skew distribution of absolute biases (see Table 4 in the
Supplementary Materials).

The second column of Table 2 displays the relative impact of PSB on meta-
analytic estimate via the overestimation factor. On average, economics meta-
analyses are, relatively, the most PSB exaggerated, inflating effect sizes by
over two times; this corroborates a prior survey on power and bias [10].
Effect sizes in environmental sciences and medical meta-analyses show smaller
yet notable relative effect size inflation. Finally, effect sizes in
psychological meta-analyses are the least inflated with the average effect
size exaggerated by 40%. In each field, the distribution of absolute biases is
right-skewed; consequently, the per meta-analysis overestimation factor
_median_ is lower than the mean (see Table 4 in the Supplementary Materials).
The median overestimation factor is relatively stable/decreasing with the
increasing number of effect size estimates per meta-analysis, suggesting that
the number of meta-analyses does not play a role in the relative size of PSB.

###  3.4 Evidence for Publication Selection Bias

Figure 2: Median, Interquartile Range, and Distribution of Posterior
Probability for the Presence of Publication Selection Bias in Each Field

Note. The width of grey area indicates density, the light grey area indicates
the interquartile range, and the black line indicates the median. The $y$-axis
is scaled according to posterior probabilities, assuming equal prior
probabilities of presence vs. absence of the publication selection bias. See
the secondary $y$-axis for Bayes factors in favor of the publication selection
bias that are independent of the assumed prior probability of the publication
selection bias.

Figure 2 shows medians, interquartile ranges, and distributions of the
posterior probability of the PSB in each field. We find the most evidence for
PSB in economics, where the typical evidence of the presence of publication
bias is moderate, median $\text{BF}_{\text{PSB}}=7.27$ corresponding to
$87.9$% posterior probability of publication selection bias. Meta-analyses in
environmental sciences and psychology have weak evidence in favor of PSB; even
though the proportion of meta-analyses showing at least moderate evidence for
PSB is still considerable (32.2% and 27.4% respectively). Meta-analyses in
medicine show the lowest proportion of at least moderate evidence in favor of
PSB (12.9%). However, the proportion of meta-analyses consistent with the
evidence of absence of PSB (i.e., $\text{BF}_{\text{PSB}}<1/3$) is also the
lowest in medicine (2.6%), indicating that the majority of medical meta-
analyses is not informative enough to provide compelling evidence for or
against publication bias. The proportion of meta-analyses with at least
moderate evidence against PSB is somewhat higher in psychology (7.1%),
economics (12.2%), and environmental sciences (12.6%). Meta-analyses with a
larger number of effect size estimates present slightly more evidence in favor
of the PSB; however, the overall disparity between the fields remains when
comparing meta-analyses with a matched number of effect size estimates (Table
17 in Supplementary Materials).

##  4 Concluding Comments

We present a comprehensive assessment of publication selection bias and its
effects on meta-analyses across medicine, environmental sciences, psychology,
and economics. Novel methods and measures allowed us to quantify the evidence
for the absence or presence of the mean effect and publication selection bias,
as well as inflation of the evidence of the effect due to the publication
selection bias. Furthermore, we estimated the bias and overestimation factor
of the effect sizes of average estimates included in meta-analyses.

Our analysis is based on all effect size estimates found in these meta-
analyses, regardless of the type outcome or how they were analyzed. One can
classify outcomes into three categories. First, some outcomes may have been
pre-specified as being of primary interest to show a desirable effect (e.g.,
the effectiveness of a medication in reducing the risk of death). Second, some
other outcomes are not pre-specified but may still be used to demonstrate some
preferred outcome; thus, they may have larger analytical flexibility (e.g.,
using alternative measures of effectiveness) and thereby are potentially more
affected by publication selection bias. Third, still other outcomes may have
been collected and analyzed without any strong interest to show some
significant result, or even with some incentive to show non-significant
results (e.g., outcomes on collected adverse events). Publication selection
bias is expected mostly in the second category, while it may be less in the
first category [38] and may be entirely absent in the third category.

Furthermore, we assumed independence of the reported primary estimates within
and between meta-analyses; that is, each reported estimate is regarded to
provide the same amount of new information as every other reported estimate.
However, estimates may be dependent between meta-analyses, e.g., a single
estimate might be used across multiple meta-analyses, and within meta-
analyses, e.g., multiple estimates obtained from a single study/primary data
set. As our data does not allow us to tackle those dependencies directly, we
discuss how each independence violation might affect the results. The between
meta-analysis dependency of estimates is of lesser importance as our
inferences are concerned with the population of meta-analyses. Consequently,
between meta-analysis dependency of estimates would only affect descriptive
summaries of the estimates themselves. The within meta-analysis dependency of
estimates is more problematic and can lead to 1) the overestimation of the
strength of evidence, as the same primary data set is conditioned upon more
than once, and 2) placing more weight on studies with multiple estimates. The
first issue is partially mitigated via the standardized evidence inflation
factor, which assesses the average evidence contribution of an estimate, i.e.,
adjusting for the number of data sets conditioned upon. However, the absolute
measures of evidence (i.e., evidence for the presence of the effect before and
after publication bias adjustment or the evidence for publication bias) can be
susceptible to overestimation, particularly in fields with relatively large
within meta-analysis dependencies such as economics or environmental science
(but see Supplementary Materials for comparison of meta-analyses with a
matched number of effect size estimates). The second issue cannot be directly
addressed; however, all presented measures are based on comparisons of two
sets of models, both of which should be affected to a similar degree, thus
hopefully canceling most of the bias that is generated by overweighting
studies with multiple estimates. Overall, we cannot exclude that the observed
between-field differences may at least partially result from systematic
differences in how meta-analyses themselves are conducted.

The milder publication selection bias in medical meta-analyses corroborates
previous findings and might have multiple concurring explanations [9, 16, 19].
First, as in other disciplines, a large share of those medical meta-analyses
with seemingly strong evidence no longer had strong evidence when PSB
adjustment was made. However, a much lower proportion of medical meta-analyses
showed strong evidence of an effect compared to the other disciplines.
Therefore, the difference between medicine and the other disciplines might be
explained by the higher proportion of meta-analyses in medicine that showed
weak evidence for an effect already before adjusting for publication selection
bias. Second, medical studies may measure phenomena that are simpler and more
stable, using methods that are more solidly and universally codified, which
reduces researchers’ “degrees of freedom” in generating and publishing
evidence [9, 16]. Third, it is also possible that the milder publication
selection bias seen in medical meta-analyses is reflecting a larger share of
meta-analyses that belonged to a category of outcomes with less pressure for
publication selection bias. Finally, medical research makes wider use of
research integrity practices, such as clinical trial registration, which might
reduce the risk of publication selection bias [39]. Perhaps, medical research
is, therefore, typically of a higher methodological quality and less subject
to bias [9].

In this paper, we documented the considerable impact of publication selection
bias on meta-analyses in a variety of disciplines. Even though we can probe
the footprint of these biases with the statistical techniques employed here,
science ultimately needs to progress toward mitigating publication bias
already while conducting and publishing the research. While the specific
patterns of researchers’ “degrees of freedom” and causes of publication
selection bias are likely to vary widely across fields, our results suggest
that the social sciences might especially benefit from adopting practices to
mitigate these, including: preregistration, greater transparency, and
registered reports [40, 41, 42].

## Highlights

\- Publication selection bias, where studies with significant or positive
results are more likely to be reported and published, distorts the available
scientific record.
\- This study surveyed over 68,000 meta-analyses from medicine, environmental
sciences, psychology, and economics to assess the extent of publication
selection bias. As a result, it underscores the importance of addressing
publication bias in evidence synthesis
\- Results suggest that meta-analyses in economics are the most affected by
publication selection bias, followed by environmental sciences and psychology.
In contrast, meta-analyses in medicine are suggested to be the least affected.
Yet, notable biases are found across all of these scientific disciplines.
\- This study documents the potential extent of publication bias in different
fields, which could help researchers and the public better understand the
limitations of research and the potential biases of research synthesis.

## Availability of Data and Materials

See https://osf.io/bgfzp/ for data and analysis scripts.

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