The Strong Factorial Conjecture

In this paper, we present an unexpected link between the Factorial Conjecture [8] and Furterʼs Rigidity Conjecture [13]. The Factorial Conjecture in dimension m asserts that if a polynomial f in m variables Xi over C is such that L(fk)=0 for all k⩾1, then f=0, where L is the C-linear map from C[X1,…,Xm] to C defined by L(X1l1⋯Xmlm)=l1!⋯lm!. 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 X2, is equal to X if m consecutive coefficients of the formal inverse (for the composition) of a(X) are zero.