An approach to the Jacobian Conjecture in terms of irreducibility and square-freeness

We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements.


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 square-free 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].
Let k be a field of characteristic zero. By k [x 1 , . . . , x n ] we denote the k-algebra of polynomials in n variables. Given polynomials f 1 , . . . , f n ∈ k [x 1 , . . . , The Jacobian Conjecture was stated by Keller in 1939 for polynomials with integer coefficients [26]. For arbitrary field k of characteristic zero it asserts the following: It is known [9] that formulations of the Jacobian Conjecture for various fields of characteristic zero (as well as for Z) are equivalent to each other. The conjecture can be expressed in terms of k-endomorphisms of the polynomial ring k [x 1 , . . . , x n ]: If a k-endomorphism ϕ of k [x 1 , . . . , x n ] satisfies the Jacobian condition jac(ϕ(x 1 ), . . . , ϕ(x n )) ∈ k \{0}, then it is an automorphism.
For more information on the Jacobian Conjecture we refer the reader to the van den Essen's book [11]. 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 k-endomorphism of k [x 1 , . . . , x n ] over a field k of characteristic zero maps variables to variables, then it is an automorphism.
A positive solution of this problem was obtained by Jelonek [24,25]. In 2006, Bakalarski proved an analogous fact for irreducible polynomials over 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 k-endomorphisms of k [x 1 , . . . , x n ] satisfying the Jacobian condition as those mapping irreducible polynomials to square-free polynomials [20,Theorem 5.1]. This fact has been further generalized by de Bondt and Yan: they proved that mapping squarefree polynomials to square-free ones is also equivalent to the Jacobian condition [7, We present our generalization of the Jacobian Conjecture for r polynomials Recall that an element a ∈ R is called square-free if it cannot be presented in the form a = b 2 c, where b, c ∈ R and b is non-invertible. 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 square-free elements of a subring are square-free 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.

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 f ∈ C[x, y], let g ∈ C[x, y] be an irreducible non-constant common factor of ∂ f /∂ x and ∂ f /∂ y. Then there exists c ∈ C such that g divides f + c.
The assertion of the above statement can be strengthened in a way that if g is irreducible, then

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 k
They noted [12,Remark 2.4] that the assumption "k is algebraically closed" cannot be dropped: The idea of a generalization (in [18]) to arbitrary field k of characteristic zero was to consider, instead of f − c, a polynomial w( f ), where w(T ) is irreducible. In the mentioned example w(T ) = T 2 + 4 since g | f 2 + 4. In fact, Freudenburg's Lemma was generalized to the case when the coefficient ring is a UFD of arbitrary characteristic.
As a consequence we obtain (see [18,Proposition 3.3]) that if k is an arbitrary field of characteristic zero, f, g ∈ k [x 1 , . . . , x n ] and g is irreducible, then A generalization of Freudenburg's Lemma to an arbitrary number of polynomials over a field of characteristic zero was obtained in [23]. Denote by jac The proof is based on the methods of proofs of earlier special cases: [ 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 p-bases of rings of constants with respect to polynomial derivations.

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.
In this way we obtain a new equivalent formulation of the Jacobian Conjecture for an arbitrary field k of characteristic zero: Every k-endomorphism of k [x 1 , . . . , x n ] mapping square-free polynomials to square-free polynomials is an automorphism.
There is a natural question if there exists a non-trivial example of an endomorphism satisfying condition (ii) such that ϕ(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].

A generalization of the Jacobian Conjecture
In [23], we generalized the Jacobian Conjecture in the following way (recall that jac Recall that by Nowicki's characterization the above assertion means that R is a ring of constants of some k-derivation of k [x 1 , . . . , x n ] (see [28,Theorem 5.4]

Analogs of Jacobian conditions for subrings
In this section we present equivalent versions of the generalized Jacobian condition from Conjecture JC(r, n, k) in terms of irreducible elements as well as square-free elements. It is useful to introduce (following [19]) the notion of a "differential gcd" for r polynomials f 1 , . . . , f r ∈ k [x 1 , . . . , x n ], where r ∈ {1, . . . , n}: The next theorem is a consequence of Theorem 2.4.
Note that under the assumptions of the above theorem, a polynomial w ∈ k [x 1 , . . . , x r ] is irreducible (square-free) if and only if w( f 1 , . . . , f r ) is an irreducible (square-free) element of k [ f 1 , . . . , f r ]. This allows us to express the above conditions in terms of the sets of irreducible elements (Irr ) and square-free elements (Sqf ) of the respective rings.
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 JC(r, n, k) motivated us to state the following question [23, Section 3].

Under what conditions is R algebraically closed in A?
In particular, the ordinary Jacobian Conjecture for r = n, A = k [x 1 , . . . , x n ], where char k = 0, asserts that if f 1 , . . . , f n ∈ A are algebraically independent over k, In order to understand a more general context of the conditions Irr R ⊂ Sqf A and Sqf R ⊂ Sqf A, when R is a subring of a domain A, we can inscribe them into the following diagram of implications [22,Proposition 3.3].
By Prime R we have denoted the set of all prime elements of R, by Gpr R the set of (single) generators of principal radical ideals of R, and by Rdl R (following [6, p. 67]) the set of radical ideals of R.

Factorial properties
Now we discuss factorial properties connected with inclusions Irr R ⊂ Irr A and Sqf R ⊂ Sqf A, where R is a subring of a unique factorization domain A.
Recall that a subring R of a domain A such that for every x, y ∈ A: is called factorially closed. Rings of constants of locally nilpotent derivations in domains of characteristic zero are factorially closed (see [10,15] for details). Note that according only to the multiplicative structure, a submonoid of a (commutative cancellative) monoid satisfying the above condition is called divisor-closed [16].
If A is a UFD, then a subring R of A that fulfills condition (ii) of Theorem 6.1 we will call square-factorially closed in A. Condition (ii) has an advantage over condition (i) since it does not involve square-free elements of R. For example, one can define the square-factorial closure of a subring R in A as an intersection of all square-factorially closed subrings of A containing R.
There arise two questions concerning the condition Irr R ⊂ Sqf A in the case when A is a UFD. Firstly, is it equivalent to Sqf R ⊂ Sqf A under some natural assumptions (like R * = A * )? If such equivalence does not hold in general, can the condition Irr R ⊂ Sqf A be expressed in a form of factoriality, similarly to the above theorem?
The notion of square-factorial closedness is relevant to the thoroughly studied notion of root closedness. Recall that a subring R of a ring A is called root closed in A if the following implication: x n ∈ R ⇒ x ∈ R holds for every x ∈ A and n 1. An interesting task would be to investigate whether square-factorial 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 R[X ; ], where is torsionless grading monoid. If for square-factorial 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 R 0 ∩ A = R (see [3]) as in Theorem 6.2. Another prospect for further research is to obtain relationships (similarly to Theorem 6.2) of square-factorial closedness with other notions, such as seminormality.