Constructing elusive functions with help of evaluation mappings

TL;DR

This paper introduces a novel method for constructing elusive functions using algebraic geometry and commutative algebra, enabling the creation of explicit examples with large circuit size and algebraic number coefficients.

Contribution

It develops a new approach based on elusive subsets and evaluation mappings, combining elimination theory and algebraic number theory to construct concrete elusive functions.

Findings

Constructed explicit elusive functions with algebraic number coefficients.

Demonstrated functions with large circuit complexity.

Provided a framework for future algebraic complexity research.

Abstract

We develop a method to construct elusive functions using techniques of commutative algebra and algebraic geometry. The key notions of this method are elusive subsets and evaluation mappings. We also develop the effective elimination theory combined with algebraic number field theory in order to construct concrete points outside the image of a polynomial mapping. Using the developed methods, for $F = C \text{or} F = R$, we construct examples of $(s,r)$-elusive functions whose monomial coefficients are algebraic numbers, which give polynomials with algebraic number coefficients of large circuit size.