A finiteness result towards the Casas-Alvero Conjecture

Autor:	Ghosh, Soham
Rok vydání:	2024
Předmět:	Mathematics - Algebraic Geometry Mathematics - Commutative Algebra 12E05 13C15 13N15 14G05 14M10 13N15
Druh dokumentu:	Working Paper
Popis:	The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The conjecture for polynomials of a fixed degree is equivalent to the projective variety of such polynomials being one-dimensional. In this paper, we show that for any algebraically closed field of arbitrary characteristic, this variety is at most two-dimensional for all positive degrees. Consequently, we show that the associated arithmetic Casas-Alvero scheme in any positive degree has finitely many rational points over any field. Along the way, we prove several rigidity results towards the conjecture. Comment: Gap in proof of Theorem A fixed. Proposition 5.3 and Corollary 5.9 added. Comments welcome
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2402.18717 Zobrazit plný text záznamu View this record from Arxiv