Abstract
This paper introduces a relation between resultant and the Jacobian determinant by generalizing Sakkalis theorem from two polynomials in two variables to the case of (n) polynomials in (n) variables. This leads us to study the results of the type: , and use this relation to attack the Jacobian problem. The last section shows our contribution to proving the conjecture.
Highlights
The Jacobian Conjecture can be stated as follows: For any integer n ≥ 1 and polynomials f1, ... , fn ∈ C[X]; the polynomial map F = (f1, ... , fn): Cn → Cn is an automorphism if det JF is a nonzero constant.Notation Throughout this paper JF is used for Jacobian matrix, and X is used to denote the variables x1, ... , xn ki is put to be the degxi(Ri(xi, u1, . . , un))
This paper introduces a relation between resultant and the Jacobian determinant by generalizing Sakkalis theorem from two polynomials in two variables to the case of (n) polynomials in (n) variables
∂Bn ∂un (xn the map F = (f, g), in a way to construct an ideal. He considers the two relative resultants polynomials of the map F. The key theorem he proved is that: The Jacobian Conjecture is true for F if and only if the leading coefficients of these two resultants belong to the Jacobian ideal
Summary
If F is bijective, the inverse will be automatically polynomial (Theorem2.1,1). It is well-known that the invertibility of F implies the invertibility of JF. In 1993, Takis Sakkalis, in his paper[3], investigated a certain connection between the zeros of a polynomial F: C2 → C2 and the Jacobian determinant of f and g (where f and g are nonzero polynomials in C[x, y]). Researches in this field are still ongoing; for example, a recent study was published in May 2020.
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