Variable lemma for polynomial rings

Abstract

Let \(k\) be a field of characteristic zero, and let \(k[x_1,\ldots,x_n]\) be a polynomial ring in \(n\) variables, where \(n\geq 3\) is an arbitrary positive integer. Assume that we have \(l\in\mathbb{N}\) algebraically independent polynomials \(F_1,\ldots,F_l\in k[x_1,\ldots,x_n]\) with \(2\leq l <n.\) In this paper, we prove that if linear parts of polynomials \(F_1,\ldots,F_l\) are linearly independent and depend only on variables \(x_1,\ldots,x_l\) and the polynomials \(F_1,\ldots,F_l\) meet some weighted-differential criteria then actually \(F_1,\ldots,F_l\in k[x_1,\ldots,x_l].\)