The Jacobian conjecture can be reduced to the consideration of polynomial maps F:Cn→Cn of the special form F=X−H, where X is the identity map and H contains only higher order terms (i.e., terms of degree at least 2). In that case, it asserts that if |J(F)|, the Jacobian determinant of F, is identically 1, then F is bijective, with a polynomial inverse. The introduction contains a convenient summary of known reductions to yet more special forms and known partial results for those forms. The reductions are formulated as families of conjectures for each pair (n,d), where n≥2 is the dimension and d≥2 is a degree bound. The five cases discussed are: each Hi has degree at most d; each Hi is homogeneous of degree d (or zero); each Hi is of the form Ldi (with Li a linear form) [degree d Drużkowski form]; the Jacobian matrix J(H) is symmetric; and, finally, each Hi is homogeneous of degree d and J(H) is symmetric. In all these five cases, the stated conjectures are equivalent to the Jacobian conjecture if they are true for a fixed d≥3 and all values of n≥2. The generic formal map of special form F=X−H in dimension n has (new) indeterminates as coefficients of the monomials (of all degrees) of H. Its formal inverse has special form, with coefficients that are polynomials in the coefficients of the generic formal map. Previous work has shown that there are explicit formulas for the coefficients of the inverse as sums of monomials indexed by isomorphism classes of certain finite trees. In the five cases above, the generic formal map can be specialized; for instance, for the case of a symmetric J(H), one can write F=X−∇P, where P is a potential function in n variables of degree d+1, and then the inverse coefficients are polynomials in the (indeterminate) coefficients of P. The individual conjectures for a given case and fixed n and d then hold if the inverse coefficients of monomials of sufficiently high degree belong to the radical of an appropriate ideal; the ideals in question are those generated by the coefficients of the monomials of 1−|J(H)| (the Jacobian ideal) or the ideal generated by the coefficients of the monomials of J(H)n (the nilpotency ideal—used in homogeneous cases). Membership and ideals are considered in the Q-algebra generated by the finitely many (because (n,d) is fixed) indeterminates of the case specific generic formal map. Of particular interest are specialized formulas for the formal inverse, and the determination of cases in which the use of the radical can be dropped. The resulting plethora of ideal membership questions are answered in some specific cases by use of a computer algebra system (details not shown). Some explicit case specific formulas are cited or developed for inverse coefficients, homogeneous components of the inverse, and generators for ideals to whose radicals inverse coefficients must belong. Bounds on the degrees of monomials for which inverse coefficient ideal membership must actually be checked are improved. As a sample of results, the last section states that if F=X−H has H homogeneous of degree d, with J(H) symmetric and J(H)3=0, then F is invertible and the degree of its inverse is less than 2d (independent of n).
Wright, David, "The Jacobian conjecture: ideal membership questions and recent advances" (2005). Mathematics Faculty Publications. 6.