Arithmetic Circuits: A Survey of Recent Results and Open Questions (Foundations and Trends(r) in Theoretical Computer Science)

Algebraic complexity theory studies the inherent difficulty of algebraic problems by quantifying the minimal amount of resources required to solve them. The most fundamental questions in algebraic complexity are related to the complexity of arithmetic circuits: providing efficient algorithms for algebraic problems, proving lower bounds on the size and depth of arithmetic circuits, giving efficient deterministic algorithms for polynomial identity testing, and finding efficient reconstruction algorithms for polynomials computed by arithmetic circuits. Arithmetic Circuits: A Survey of Recent Results and Open Questions surveys the field of arithmetic circuit complexity. It covers the main results and techniques in the area, with an emphasis on works from the last two decades. In particular, it discusses the classical structural results including VP = VNC2 and the recent developments highlighting the importance of depth-4 circuits, the classical lower bounds of Strassen and Baur-Strassen and the recent lower bounds for multilinear circuits and formulas, the advances made in the area of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. It also presents many open questions that may be considered as natural "next steps" given the current state of knowledge.