Trinomial containment in polynomial ideals is undecidable

Abstract

We prove that deciding whether an ideal in a polynomial ring contains a trinomial is impossible on a Turing machine. More precisely, from an integer polynomial $P$ we compute generators of an ideal $I_P$ in a polynomial ring over $\mathbb{Q}$ such that $I_P$ contains a trinomial if and only if $P$ has an integral zero. By the MRDP theorem this problem is undecidable. A universal halting polynomial gives a computable family of ideals in one fixed polynomial ring, with uniform bounds on colength, generator count, and generator degree, for which the containment of a trinomial encodes the halting problem.