Bibliographic Information: Balaji, N., Nosan, K., Shirmohammadi, M., & Worrell, J. (2024). Identity Testing for Radical Expressions. [Publication Information to be added upon publication]
Research Objective: This paper investigates the computational complexity of the Radical Identity Testing (RIT) problem, which involves determining whether a polynomial expression evaluated at real radicals equals zero. The authors aim to provide more efficient algorithms and complexity bounds for RIT and its variant, 2-RIT, where inputs are square roots of distinct odd primes.
Methodology: The authors employ a symbolic approach, generalizing the fingerprinting technique used in Arithmetic Circuit Identity Testing (ACIT). They leverage concepts from algebraic and analytic number theory, including Galois theory, Chebotarev's density theorem, quadratic reciprocity, and Dirichlet's theorem on primes in arithmetic progressions.
Key Findings:
Main Conclusions: The study provides novel insights into the complexity of RIT and 2-RIT, demonstrating more efficient algorithms under GRH. The use of algebraic and analytic number theory tools highlights the interplay between these fields and computational complexity.
Significance: This research contributes to our understanding of identity testing problems, a fundamental area in computational complexity with broad applications in algorithm design and other areas of computer science.
Limitations and Future Research: The coNP upper bound for RIT relies on GRH. Future work could explore unconditional complexity bounds or investigate the possibility of derandomizing the 2-RIT algorithm. Additionally, exploring extensions to more general radical expressions or other classes of algebraic numbers could be of interest.
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