The Strong Factorial Conjecture

In this paper, we present an unexpected link between the Factorial Conjecture ([8]) and Furter’s 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(f) = 0 for all k ≥ 1, then f = 0, where L is the C-linear map from C[X1, · · · , Xm] to C defined by L(X l1 1 · · ·X lm m ) = 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 X, is equal to X if m consecutive coefficients of the formal inverse (for the composition) of a(X) are zero. 1. Presentation In Section 2, we recall the Factorial Conjecture from [8]. We give a natural stronger version of this conjecture which gives the title of this paper. We also recall the Rigidity Conjecture from [13]. We present an additive and a multiplicative inversion formula. We use the multiplicative one to prove that the Rigidity Conjecture is a very particular case of the Strong Factorial Conjecture (see Theorem 2.25). As an easy corollary we obtain a new case of the Factorial Conjecture (see Corollary 2.28). In section 3, we study the Strong Factorial Conjecture in dimension 2. We give a new proof of the Rigidity Conjecture R(2) (see Subsection 3.1) using the Zeilberger Algorithm (see [16]). We study the case of two monomials (see Subsection 3.2). In Section 4 (resp. 5) we shortly give some historical details about the origin of the Factorial Conjecture (resp. the Rigidity Conjecture). 2. The bridge In this section, we fix a positive integer m ∈ N+. By C = C[X1, . . . , Xm], we denote the C-algebra of polynomials in m variables over C. Email addresses: [email protected] (Eric Edo), [email protected] (Arno van den Essen) Preprint submitted to Elsevier May 28, 2013 2.1. The Strong Factorial Conjecture We recall the definition of the factorial map (see [8] Definition 1.2): Definition 2.1. We denote by L : C → C the linear map defined by L(X l1 1 . . . X lm m ) = l1! . . . lm! for all l1, . . . lm ∈ N. Remark 2.2. Let σ ∈ Sm be a permutation of the set {X1, . . . , Xm}. If we extend σ to an automorphism σ̃ of the C-algebra C, then for all polynomials f ∈ C, we have L(σ̃(f)) = L(f). Remark 2.3. The linear map L is not compatible with the multiplication. Nevertheless, L(fg) = L(f)L(g) if f, g ∈ C are two polynomials such that there exists an I ⊂ {1, . . . ,m} such that f ∈ C[Xi ; i ∈ I] and g ∈ C[Xi ; i 6∈ I]. We recall the Factorial Conjecture (see [8] Conjecture 4.2). Conjecture 2.4 (Factorial Conjecture FC(m)). For all f ∈ C, (∀k ∈ N+)L(f ) = 0 ⇒ f = 0. To state some partial results about this conjecture it is convenient to introduce the following notation: Definition 2.5. We define the factorial set as the following subset of C: F = {f ∈ C r {0} ; (∃k ∈ N+)L(f ) 6= 0} ∪ {0}. Remark 2.6. Let f ∈ C be a polynomial, we have f ∈ F [m] if and only if: (∀k ∈ N+)L(f ) = 0 ⇒ f = 0. In other words, the factorial set F [m] is the set of all polynomials satisfying the Factorial Conjecture FC(m) and this conjecture is equivalent to F = C. To give a stronger version of this conjecture we introduce the following subsets of C: Definition 2.7. For all n ∈ N+, we consider the following subset of C: F n = {f ∈ C [m] r {0} ; (∃k ∈ {n, . . . , n+N (f)− 1})L(f) 6= 0} ∪ {0} where N (f) denotes the number of (nonzero) monomials in f . We define the strong factorial set as: F [m] ∩ = ⋂