An approach to the Jacobian Conjecture in terms of irreducibility and squarefreeness
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Abstract
We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and squarefree elements.
Keywords
Jacobian conjecture Keller map Irreducible element Squarefree element Factorial propertyMathematics Subject Classification
13F20 14R15 13F151 Introduction
The Jacobian Conjecture is one of the most important open problems stimulating modern mathematical research [29]. Its long history is full of equivalent formulations and wrong proofs. In this article we give a survey of a new purely algebraic approach to the Jacobian Conjecture in terms of irreducible elements and squarefree elements, based mainly on: the authors’ paper [20], de Bondt and Yan’s paper [7], authors’ paper [23], and the joint paper with Matysiak [22].
It is known [9] that formulations of the Jacobian Conjecture for various fields of characteristic zero (as well as for \({\mathbb {Z}}\)) are equivalent to each other. The conjecture can be expressed in terms of kendomorphisms of the polynomial ring Open image in new window :If polynomials Open image in new window satisfy the Jacobian condition Open image in new window , then Open image in new window .
For more information on the Jacobian Conjecture we refer the reader to the van den Essen’s book [11].If a kendomorphism \(\varphi \) of Open image in new window satisfies the Jacobian condition Open image in new window , then it is an automorphism.
A primary motivation of our approach can be found in a question of van den Essen and Shpilrain from 1997 [13, Problem 1], whether if a kendomorphism of Open image in new window over a field k of characteristic zero maps variables to variables, then it is an automorphism. A polynomial \(f\in k\) is called a variable if there exist polynomials Open image in new window such that Open image in new window . A positive solution of this problem was obtained by Jelonek [24, 25]. In 2006, Bakalarski proved an analogous fact for irreducible polynomials over \(\mathbb {C}\) ([5, Theorem 3.7], see also [1]). Namely, he proved that a complex polynomial endomorphism is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. One of the authors in 2013 obtained a characterization of kendomorphisms of Open image in new window satisfying the Jacobian condition as those mapping irreducible polynomials to squarefree polynomials [20, Theorem 5.1]. This fact has been further generalized by de Bondt and Yan: they proved that mapping squarefree polynomials to squarefree ones is also equivalent to the Jacobian condition [7, Corollary 2.2].
We present our generalization of the Jacobian Conjecture for r polynomials Open image in new window , where k is a field of characteristic zero and \(r\leqslant n\): if all jacobians (with respect to r variables) are relatively prime, then Open image in new window is algebraically closed in Open image in new window [23]. Next we present equivalent versions of this generalized Jacobian condition in terms of the mentioned ksubalgebra: all irreducible (resp. squarefree) elements of Open image in new window are squarefree in Open image in new window [23, Theorem 2.4]. Recall that an element \(a\in R\) is called squarefree if it cannot be presented in the form \(a=b^2c\), where \(b,c\in R\) and b is noninvertible. It is reasonable to consider such properties in a general case, e.g. for subrings of unique factorization domains. In this case the property that squarefree elements of a subring are squarefree in the whole ring can be expressed in some factorial form [23, Theorem 3.4]. At the end we discuss possible directions of future research.
2 Freudenburg’s Lemma and its generalizations
A motivation for the main preparatory fact (Theorem 2.4) comes from generalizations of the following lemma of Freudenburg from [14].
Theorem 2.1
(Freudenburg’s Lemma) Given a polynomial Open image in new window , let Open image in new window be an irreducible nonconstant common factor of \({\partial f}/{\partial x}\) and \({\partial f}/{\partial y}\). Then there exists \(c\in \mathbb {C}\) such that g divides \(f+c\).
Theorem 2.2
(van den Essen, Nowicki, Tyc) Let k be an algebraically closed field of characteristic zero. Let Q be a prime ideal of the ring Open image in new window and Open image in new window . If for each i the partial derivative \({\partial f}/{\partial x_i}\) belongs to Q, then there exists \(c\in k\) such that \(fc\in Q\).
They noted [12, Remark 2.4] that the assumption “k is algebraically closed” cannot be dropped: for \(f=x^3+3x\) and \(Q=(g)\), where \(g=x^2+1\), in Open image in new window we have Open image in new window , but Open image in new window for any \(c\in {\mathbb {R}}\). The idea of a generalization (in [18]) to arbitrary field k of characteristic zero was to consider, instead of \(fc\), a polynomial w(f), where w(T) is irreducible. In the mentioned example \(w(T)=T^2+4\) since Open image in new window . In fact, Freudenburg’s Lemma was generalized to the case when the coefficient ring is a UFD of arbitrary characteristic.
Theorem 2.3
 (a)
If \(\mathrm{char\,}K=0\), then there exists an irreducible polynomial Open image in new window such that \(w(f)\in Q\).
 (b)
If \(\mathrm{char\,}K=p>0\), then there exist Open image in new window such that Open image in new window , \(b\not \in Q\) and \(bf+c\in Q\).
Theorem 2.4
 (i)
Open image in new window for every Open image in new window ,
 (ii)
Open image in new window for some irreducible polynomial Open image in new window ,
 (iii)
Open image in new window for some squarefree polynomial Open image in new window .
The proof is based on the methods of proofs of earlier special cases: [20, Theorem 4.1] and de Bondt and Yan’s [7, Theorem 2.1].
Note also that a positive characteristic analog of Freudenburg’s Lemma for r polynomials in n variables was obtained in [21]. It was connected with a characterization of pbases of rings of constants with respect to polynomial derivations.
3 A characterization of Keller maps
In this section we present the main result of [20] and its substantial extension by de Bondt and Yan from [7]. Note the following consequence of Theorem 2.4 in the case \(r=n\).
Theorem 3.1
 (i)
 (ii)
the polynomial \(w(f_1,\dots ,f_n)\) is squarefree for every irreducible polynomial Open image in new window ,
 (iii)
the polynomial \(w(f_1,\dots ,f_n)\) is squarefree for every squarefree polynomial Open image in new window .
The above equivalence can be expressed as a characterization of endomorphisms satisfying the Jacobian condition analogous to the characterization of automorphisms from Bakalarski’s theorem [5, Theorem 3.7].
Theorem 3.2
 (i)
 (ii)
for every irreducible polynomial Open image in new window the polynomial \(\varphi (w)\) is squarefree,
 (iii)
for every squarefree polynomial Open image in new window the polynomial \(\varphi (w)\) is squarefree.
There is a natural question if there exists a nontrivial example of an endomorphism satisfying condition (ii) such that \(\varphi (w)\) is reducible for some irreducible w. An affirmative answer to this question is equivalent to the negation of the Jacobian Conjecture [20, Section 6, Remark 1].Every kendomorphism of Open image in new window mapping squarefree polynomials to squarefree polynomials is an automorphism.
4 A generalization of the Jacobian Conjecture
In [23], we generalized the Jacobian Conjecture in the following way (recall that Open image in new window denotes the Jacobian determinant of the polynomials \(f_1,\ldots ,f_r\) with respect to Open image in new window ).
Conjecture
Recall that by Nowicki’s characterization the above assertion means that R is a ring of constants of some kderivation of Open image in new window (see [28, Theorem 5.4], [27, Theorem 4.1.4], [10, 1.4]).

Open image in new window implies the ordinary Jacobian Conjecture for r polynomials in r variables over k [23, Lemma 1.1],

Open image in new window is true ([4, Proposition 14], see also [17, a remark before Proposition 4.2]),

the reverse implication in Open image in new window need not to be true if \(r<n\), we may take for example \(f_1=x_1^2x_2\), \(f_2=x_3\), \(\dots \), \(f_r=x_{r+1}\) [23, Remark 1.2].
5 Analogs of Jacobian conditions for subrings
Theorem 5.1
 (i)
 (ii)
for every irreducible polynomial Open image in new window the polynomial \(w(f_1,\dots ,f_r)\) is squarefree,
 (iii)
for every squarefree polynomial Open image in new window the polynomial \(w(f_1,\dots ,f_r)\) is squarefree.
Note that under the assumptions of the above theorem, a polynomial Open image in new window is irreducible (squarefree) if and only if \(w(f_1,\ldots ,f_r)\) is an irreducible (squarefree) element of Open image in new window . This allows us to express the above conditions in terms of the sets of irreducible elements (\(\mathrm{Irr\,}\)) and squarefree elements (\(\mathrm{Sqf\,}\)) of the respective rings.
Theorem 5.2
 (i)
 (ii)
\(\mathrm{Irr\,}R\subset \mathrm{Sqf\,}A\),
 (iii)
\(\mathrm{Sqf\,}R\subset \mathrm{Sqf\,}A\).
Therefore we may consider conditions (ii) and (iii) in a general case, when A is a domain (a commutative ring with unity without zero divisors) and R is a subring of A, and we may call them analogs of the Jacobian condition (i). Conjecture \(\text {JC}(r,n,k)\) motivated us to state the following question [23, Section 3].
A general question
In particular, the ordinary Jacobian Conjecture for \(r=n\), Open image in new window , where \(\mathrm{char\,}k=0\), asserts that if \(f_1,\dots ,f_n\in A\) are algebraically independent over k, Open image in new window and \(\mathrm{Sqf\,}R\subset \mathrm{Sqf\,}A\), then \(R=A\).
6 Factorial properties
Now we discuss factorial properties connected with inclusions \(\mathrm{Irr\,}R\subset \mathrm{Irr\,}A\) and \(\mathrm{Sqf\,}R\subset \mathrm{Sqf\,}A\), where R is a subring of a unique factorization domain A.
Under natural assumptions we can express also the condition \(\mathrm{Sqf\,}R\subset \mathrm{Sqf\,}A\) in a form of factoriality. If R is a domain, by \(R_0\) we denote its field of fractions.
Theorem 6.1
 (i)
\(\mathrm{Sqf\,}R\subset \mathrm{Sqf\,}A\),
 (ii)
for every \(x\in A\), \(y\in \mathrm{Sqf\,}A\), if Open image in new window , then \(x,y\in R\).
If A is a UFD, then a subring R of A that fulfills condition (ii) of Theorem 6.1 we will call squarefactorially closed in A. Condition (ii) has an advantage over condition (i) since it does not involve squarefree elements of R. For example, one can define the squarefactorial closure of a subring R in A as an intersection of all squarefactorially closed subrings of A containing R.
There arise two questions concerning the condition \(\mathrm{Irr\,}R\subset \mathrm{Sqf\,}A\) in the case when A is a UFD. Firstly, is it equivalent to \(\mathrm{Sqf\,}R\subset \mathrm{Sqf\,}A\) under some natural assumptions (like Open image in new window )? If such equivalence does not hold in general, can the condition \(\mathrm{Irr\,}R\subset \mathrm{Sqf\,}A\) be expressed in a form of factoriality, similarly to the above theorem?
Theorem 6.2
([23, Theorem 3.6]) Let A be a unique factorization domain. Let R be a subring of A such that Open image in new window and \(R_0\cap A=R\). If R is squarefactorially closed in A, then R is root closed in A.
An interesting task would be to investigate whether squarefactorial closedness is stable under various operations and extensions. Such kind of results were obtained for example for root closedness (see [2, 3, 8, 30]). The latter is stable for instance under homogeneous grading and under passages to polynomial extension, to power series extension, to rational functions extension, to semigroup ring Open image in new window , where \({\Gamma }\) is torsionless grading monoid. If for squarefactorial closedness some property would not be valid in general, then under what additional assumptions it will? For example, stability of root closure under passage to power series extension is acquired by imposing the assumption that a subring R is von Neumann regular (see [30]) or Open image in new window (see [3]) as in Theorem 6.2. Another prospect for further research is to obtain relationships (similarly to Theorem 6.2) of squarefactorial closedness with other notions, such as seminormality.
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