Let $K$ be the field of real or $p$-adic numbers, and $F(x) = (f_1(x), . . ., f_m(x))$ be such that $1, f_1 , . . . , f_m$ are linearly independent polynomials with coefficients in $K$. In the present talk, we will prove that for the field $K$, the Borel chromatic number of of the Cayley graph of $K^m$ with respect to these polynomials is infinite. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, due to Bachoc, DeCorte, Oliveira and Vallentin, combined with an analysis of certain oscillatory integrals over local fields.