A PCP of Proximity for Real Algebraic Polynomials

Abstract

Let $f:F_q^k \mapsto {\mathbb R}$ be a function given via its table of values, where $F_q:=\{0,1,\ldots ,q-1\} \subset {\mathbb R}, k,q \in {\mathbb N}.$ We design a randomised verification procedure in the BSS model of computation that verifies if f is close to an algebraic polynomial of maximal degree $d \in {\mathbb N}$ in each of its variables. If f is such an algebraic polynomial there exists a proof certificate that the verifier will accept surely. If f has at least distance $\epsilon > 0$ to the set of max-degree algebraic polynomials on $F_q^k$, the verifier will reject any proof with probability at least $\frac{1}{2}$ for large enough q. The verification procedure establishes a real number PCP of proximity, i.e., it has access to both the values of f and the additional proof certificate via oracle calls. It uses $O(k\log {q})$ random bits and reads O(1) many components of both f and the additional proof string, which is of length $O((kq)^{O(k)}).$ The paper is a contribution to the not yet much developed area of designing PCPs of proximity in real number complexity theory.