2024/11/12 by Albert Atserias, Atserias, Albert, Iddo Tzameret +1 · 1 citation
Mathematics · #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Logic (math.LO) #Logic in Computer Science (cs.LO) #Mathematical Dynamics and Fractals #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2411.07966
openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
The Schwartz-Zippel Lemma states that if a low-degree multivariate polynomial with coefficients in a field is not zero everywhere in the field, then it has few roots on every finite subcube of the field. This fundamental fact about multivariate polynomials has found many applications in algorithms, complexity theory, coding theory, and combinatorics. We give a new proof of the lemma that offers some advantages over the standard proof. First, the new proof is more constructive than previously known proofs. For every given side-length of the cube, the proof constructs a polynomial-time computable and polynomial-time invertible surjection onto the set of roots in the cube. The domain of the surjection is tight, thus showing that the set of roots on the cube can be compressed. Second, the new proof can be formalised in Buss' bounded arithmetic theory S12 for polynomial-time reasoning. One consequence of this is that the theory S12 + dWPHP(PV) for approximate counting can prove that the problem of verifying polynomial identities (PIT) can be solved by polynomial-size circuits. The same theory can also prove the existence of small hitting sets for any explicitly described class of polynomials of polynomial degree. To complete the picture we show that the existence of such hitting sets is equivalent to the surjective weak pigeonhole principle dWPHP(PV), over the theory S12. This is a contribution to a line of research studying the reverse mathematics of computational complexity. One consequence of this is that the problem of constructing small hitting sets for such classes is complete for the class APEPP of explicit construction problems whose totality follows from the probabilistic method. This class is also known and studied as the class of Range Avoidance Problems.