The Strong Factorial Conjecture

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.