Existing polynomial identity bounds

The existing A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots says that a nonzero univariate polynomial of degree $m$ over an integral domain has at most $m$ distinct roots. A field is an integral domain: if $ab=0$ and $a\mathrm{\ne }0$, multiplication by $a^{-1}$ gives $b=0$.

The existing The Schwartz-Zippel lemma says that a nonzero formal polynomial of total degree at most $d$ over a field vanishes at a uniform point of $S^n$ with probability at most $d\mathrm{/}\mathrm{\mid }S\mathrm{\mid }$, for nonempty finite $S\subseteq F$.

Apply such bounds to a nonzero difference of formal polynomials. Individual degree bounds concern one variable at a time; total degree bounds concern sums of exponents within a monomial. Sum-check's round comparison is univariate and needs only the root bound. When a degree bound is at least the field size, the resulting probability bound may be vacuous. Distinct formal polynomials over a finite field need not define distinct functions on the whole field.