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Key research themes
1. How can geometric and lattice-based models enhance the construction and intuition of finite fields over finite fields?
This research area investigates alternative, more geometric or lattice-theoretic constructions of finite fields over finite fields, aiming to provide intuitive, visual models for field extensions. The goal is to bridge the gap between abstract algebraic definitions and geometric/topological intuition, thereby improving pedagogical approaches and enabling deeper understanding of concepts like Frobenius elements and reciprocity laws at finite field levels.
Key finding: Introduces lattice models extending the 1-dimensional lattice Z modulo prime p to higher-dimensional lattices representing finite field extensions F_{p^d}. This approach provides a geometric and analytic-topologic... Read more
Key finding: Develops combinatorial and analytic estimates involving sum-product phenomena in finite fields F_q, utilizing incidence theorems and Kloosterman sum bounds. These results facilitate quantitative understanding of additive and... Read more
Key finding: Presents a broad survey connecting finite field theory with geometric and algebraic structures such as design theory, error-correcting codes, and algebraic geometry. Highlights the importance of polynomials and finite field... Read more
2. What structural properties and group actions characterize polynomial automorphisms over finite fields, and how do these relate to permutation properties and complexity?
This research theme explores the algebraic and combinatorial structure of polynomial maps and automorphisms over finite fields, their induced permutations on finite vector spaces, and the classification into even and odd permutations. It examines fundamental problems such as the existence of ‘odd’ polynomial automorphisms in certain fields, the tame versus wild classification of automorphisms, and their applications in algebraic geometry and cryptography.
Key finding: Establishes that polynomial automorphisms over finite fields induce even permutations (elements of the alternating group) on F_q^n for q=2^m with m≥2, while for q odd or q=2, any permutation can arise from an automorphism.... Read more
Key finding: Characterizes permutation polynomials of the form F(X) = G(X) + γ Tr(H(X)) over F_{2^n}, reducing the problem to identification of Boolean functions possessing linear structures. This structural insight links polynomial... Read more
Key finding: Proves explicit bounds on the parameters of permutation binomials f(x) = a x^n + x^m over finite fields, showing constraints on the degree and connection to the gcd of exponent differences and field size. Demonstrates that... Read more
3. How can computational complexity and algebraic factorization challenges for polynomials over finite fields be classified and addressed?
Focuses on the algorithmic and complexity-theoretic challenges involved in factorization and irreducibility testing of polynomials over finite fields or integers, often with additional constraints such as factor evaluation at points or coefficient sums. This area also covers constructive methods for producing irreducible polynomials, factoring in computational algebra, and the hardness of determining factors with prescribed algebraic properties.
Key finding: Shows NP-completeness in the strong sense for deciding existence of polynomial factors over the integers with prescribed factor values at fixed points or matching coefficient sums, demonstrating that even restricted... Read more
Key finding: Provides a polynomial time (under the extended Riemann hypothesis) deterministic algorithm to find irreducible polynomials of fixed degree over F_p, utilizing explicit class field theory and cyclotomic fields. Bridges the gap... Read more
Key finding: Proves an analogue of the Bateman–Horn conjecture for irreducibility of polynomial values over large finite fields: given irreducible and separable polynomials, asymptotically counts the number of monic polynomials f such... Read more
Key finding: Develops optimized M-term Karatsuba-like algorithms that reduce multiplication complexity of binary polynomials in finite field arithmetic. Demonstrates recursive structures that minimize multiplication and addition... Read more