Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method

We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three independent key ingredients: (1) We allow asymmetry in the interpolation procedure. This allows to prove, via the usual cardinality argument, the existence of auxiliary divisors needed for the bounds, up to optimal degree. (2) We give an alternative proof for the existence of these auxiliary divisors, which is constructive, and works also in the symmetric case, although it requires the curves to have sufficiently many points. (3) We allow the method to deal not only with extensions of finite fields, but more generally with monogenous algebras over finite fields. This leads to sharper bounds, and is designed also to combine well with base field descent arguments in case the curves do not have sufficiently many points. As a main application of these techniques, we fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-Ozbudak. Besides, generalities on interpolation systems, as well as on symmetric and asymmetric bilinear complexity, are also discussed.

💡 Research Summary

The paper revisits the problem of determining the bilinear complexity μ(A/K) of multiplication in finite‑dimensional algebras A over a field K, with a particular focus on extensions of finite fields and monogenic algebras. After recalling the tensor‑rank definition of μ and its symmetric counterpart μ_sym, the author explains how the classic Chudnovsky‑Chudnovsky interpolation method yields linear‑in‑degree upper bounds for μ(F_{q^n}/F_q) by evaluating functions on algebraic curves and using auxiliary divisors.

The main contribution is a systematic generalisation of this method in three directions. First, the author allows asymmetric interpolation: instead of requiring a single auxiliary divisor D satisfying two simultaneous conditions (2D‑G zero‑dimensional and D‑G′ non‑special), the new framework introduces two divisors D₁ and D₂ that are handled independently (both D_i‑G′ non‑special, and D₁+D₂‑G zero‑dimensional). This eliminates the need for the “doubling map” in the divisor class group, which in earlier works caused a non‑injectivity error and forced an extra factor equal to the size of the 2‑torsion subgroup. By avoiding that map, the cardinality argument becomes correct and yields optimal degree g‑1 for the auxiliary divisors, where g is the genus of the curve.

Second, the paper supplies two proofs of the existence of such divisors. The first is a non‑constructive cardinality argument that works whenever the underlying curve has sufficiently many rational points; it is simple, works for the asymmetric system, and attains the optimal degree bound. The second is a constructive method based on the theory of Weierstrass gaps and order sequences, extending a technique previously developed by the author for a different context. This constructive approach also applies to the original symmetric system, but requires stronger hypotheses (e.g., many rational points or large base fields).

Third, the author expands the scope from finite‑field extensions to monogenic algebras A = F_q