Some Polynomial Maps with Jacobian Rank Two or Three

Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian Conjecture

The Jacobian Conjecture: linear triangularization for homogeneous polynomial maps in dimension three

Let k be a field of characteristic zero and F : k → k a polynomial map of the form F = x + H, where H is homogeneous of degree d ≥ 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T−1HT = (0, h2(x1), h3(x1, x2)), where the hi are homogeneous of degree d. As a consequence of this r...

Identifiability of an X-rank decomposition of polynomial maps

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the Xrank decomposition — a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of the pol...

Some Reductions on Jacobian Problem in Two Variables

Let f = (f1, f2) be a regular sequence of affine curves in C. Under some reduction conditions achieved by composing with some polynomial automorphisms of C, we show that the intersection number of curves (fi) in C 2 equals to the coefficient of the leading term x in g2, where n = deg fi (i = 1, 2) and (g1, g2) is the unique solution of the equation yJ(f) = g1f1 + g2f2 with deg gi ≤ n−1. So the ...

Inversion of polynomial species and jacobian conjecture

We consider the following problem in the theory of (virtual) species of structures: Given a polynomial species F such that its derivative dF equals 1, will its inverse w.r.t substitution F h?1i also be polynomial species? We shall see that this is in general not the case. From this result it follows that an analogue of the jacobian conjecture in the context of species does not hold.