$K$ be an infinite field. Let $A=K[Y_1,...Y_n]/(f_1,...,f_r)$ be a finitely generated $K$-algebra. Let $g \in K[Y_1,...Y_n]$. Let suppose deg $f_i=d_i$. Let $d= \prod
d_i$. Let suppose $X$= Spec $A$ thought of as a closed subscheme of $\mathbb {A}^n_K$ has a $\bar K$-valued point at which $g$ is non-vanishing (thought of as a regular function on $X$). Then there exists an extension field $L$ of $K$ with deg $[L:K] \leq d$ such that $X$ has a $L$-valued point at which $g$ is non-vanishing.

Let $V(g)\subset X$ be the closed subscheme of $X$ defined by the intersection of the vanishing locus of $g$ with $X$. Let $X'=X \backslash V(g)$ be an open affine subscheme of $X$. Now by hypothesis $X$ has a $\bar K$-valued point. By restricting to a irreducible component of Spec $A$ containing the $\bar K$ valued point, we can assume that $X$ is irreducible. Let dim $X=m$. By a linear change of coordinates, we can perform a Noether normalisation such there exists $m$ elements $t_1,...,t_m$ in $A$ such that $A$ is integral over $B=K[t_1,...,t_m]$ and the induced map $f$: Spec $A \rightarrow$ Spec $B$ on affine schemes corresponding to the inclusion of $B$ in $A$ has degree atmost $d$. 