New zero lemmas in commutative algebraic groups
The author generalizes and improves his earlier zero estimate [\textit{P. Philippon}, Bull. Soc. Math. Fr. 114, 355-383 (1986; Zbl 0617.14001) and Bull. Soc. Math. Fr. 115, 397-398 (1987; Zbl 0634.14001)]. Given a commutative algebraic group \(G\) over an algebraically closed field \(K\) and a subset of \(G(K)\), the goal is to show that a polynomial of not too large a degree cannot vanish on this set. The statement involves multiplicities. In the previous results (op. cit.), multiplicities were uniform. Here, a more general situation is considered. At the same time an improvement is introduced in the final estimate. Further related recent results are due to \textit{M. Nakamaye} [Bull. Soc. Math. Fr. 123, No. 2, 155-188 (1995; Zbl 0841.11037)] and \textit{L. Denis} [Comment. Math. Helv. 70, No. 2, 235-247 (1995; Zbl 0856.11028)].
zeros of polynomials over an algebraically closed field
scientific article; zbMATH DE number 1026404