Algebraic properties of word equations

In [2], Aleksi Saarela has introduced a new approach to word equations that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equations with minimal defect effect from quadratic to linear. 1. Multivariate polynomials Throughout the text, N0 means the set of all nonnegative integers. Q(x) is the field of fractions of the polynomial ring Z[x], and Q(X) the field of fractions of the polynomial ring Z[X] = Z[X1, X2, . . . , Xn]. By · we denote the standard inner product. Let α ∈ Z. Then (α)i is the i-th coordinate of α and α⊕, α⊖ denote the uniquely determined nonnegative vectors for which α = α⊕ − α⊖ and α⊕ · α⊖ = 0. Suppose that α ∈ N0 and denote X α = ∏n i=1X (α)i i ∈ Z[X]. If p ∈ Z[X] and Ωα : Z[X] → Z[x] is the evaluation homomorphism Ωα (p(X1, . . . , Xn)) = p(x (α)1 , . . . , xn), then we will write p(α) instead Ωα(p). Lemma 1. Let α, β ∈ N0 , c ∈ N and γ ∈ N n 0 . Then: (1) [X −X ](γ) = x ( x − 1 ) in Q(x), (2) [X −X ](γ) = 0 iff (α− β) · γ = 0, (3) X −X = (X −X) ∑c−1 i=0 X . Proof. (1) The equality is a result of direct computation: