2025/12/09 by Lindy, Etna, Noferini, Vanni
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Tensor decomposition and applications
paper · doi:10.48550/arxiv.2512.08550
openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28
Let \mathbbK be a field and let f,g ∈ \mathbbK[x,y] be such that the ideal ⟨ f,g ⟩ is zero-dimensional. We study the Sylvester and Bézout resultant polynomial matrices, built by interpreting f and g as univariate polynomials in x with coefficients in \mathbbK[y]. We characterize their Smith forms over \mathbbK[y] in terms of the dual spaces of differential operators, that were defined and studied by H. M. Möller et al. In particular, if \mathbbK is algebraically closed we show that, if the leading coefficients of f and g are coprime over \mathbbK[y], then the partial multiplicities of the Sylvester and Bézout resultant matrices coincide with certain integers, that we call Möller indices. These indices are uniquely determined by ⟨ f,g ⟩, and can be easily computed from a Gauss basis, as defined in [M. G. Marinari, H. M. Möller, T. Mora, Trans. Amer. Math. Soc. 348(8):3283--3321, 1996], of the dual spaces. We then generalize this result to the case of common factors in the leading coefficients, which correspond to intersections at x=∞, again describing all the invariant factors of Sylvester and Bézout resultant matrices. As a corollary, this fully characterizes the algebraic multiplicity of all the roots of the resultant Resx(f,g) ∈ \mathbbK[y] in terms of the intersection multiplicities for f and g, including those arising from infinite intersections. We discuss both algebraic and computational implications of our results.