Algebraic-Geometric Sieve: Using Algebraic Curves and Varieties to Tackle Additive Problems

Inspired by the cross-pollination seen in Sawin, Shusterman, and Kowalski (2022) for polynomials over finite fields, this project seeks to generalize sieve methods through the lens of algebraic geometry. Instead of sieving over the integers or finite fields directly, one considers sieving over rational points of algebraic curves or higher-dimensional varieties, using geometric invariants (like genus or intersection theory) to guide the process. This synthesis could reveal new structures in the solution sets of additive equations, particularly where traditional techniques are hampered by algebraic dependencies. The approach also has the potential to connect with recent advances in arithmetic geometry, leading to a deeper understanding of exceptional sets and uniformity in additive problems, and aligns with the “create novel syntheses” heuristic.

References:

1. BINARY ADDITIVE PROBLEMS FOR POLYNOMIALS OVER FINITE FIELDS. W. Sawin, M. Shusterman, E. Kowalski (2022).
2. BINARY ADDITIVE PROBLEMS FOR POLYNOMIALS OVER FINITE FIELDS. W. Sawin, M. Shusterman, E. Kowalski (2022).