Bence Csajbók

Intersecting families and nonvanishing multivariate polynomials over finite fields

Let $\mathcal{P}_{n,d}$ be the space of polynomials in $n$ variables over $\mathbb{F}_q$ of degree at most $d$. Two polynomials $f,g\in\mathcal{P}_{n,d}$ intersect if $f(\mathbf a)=g(\mathbf a)$ for some $\mathbf a\in\mathbb{F}_q^n$. A star consists of all polynomials $f\in\mathcal{P}_{n,d}$ satisfying $f(\mathbf a)=b$...