The Strong Factorial Conjecture

TL;DR

This paper explores a surprising connection between the Factorial Conjecture and Furter's Rigidity Conjecture, providing new insights into polynomial behavior and inverse coefficients in complex variables.

Contribution

It establishes an unexpected link between two prominent conjectures in polynomial algebra, potentially advancing understanding in algebraic geometry and polynomial automorphisms.

Findings

Linked the Factorial Conjecture with Furter's Rigidity Conjecture

Provided conditions under which polynomial maps are determined by inverse coefficients

Suggested new approaches to longstanding conjectures in polynomial theory

Abstract

In this paper we present an unexpected link between the Factorial Conjecture and Furter's Rigidity Conjecture. The Factorial Conjecture in dimension $m$ asserts that if a polynomial $f$ in $m$ variables $X_i$ over $\C$ is such that ${\cal L}(f^k)=0$ for all $k\geq 1$, then $f=0$, where ${\cal L}$ is the $\C$-linear map from $\C[X_1,...,X_m]$ to $\C$ defined by ${\cal L}(X_1^{l_1}... X_m^{l_m})=l_1!... l_m!$. The Rigidity Conjecture asserts that a univariate polynomial map $a(X)$ with complex coefficients of degree at most $m+1$ such that $a(X)=X$ mod $X^2$, is equal to $X$ if $m$ consecutive coefficients of the formal inverse of $a(X)$ are zero.