Constructing elusive functions with help of evaluation mappings

We develop a method to construct elusive functions using techniques of commutative algebra and algebraic geometry. The key notions of this method are elusive subsets and evaluation mappings. We also develop the effective elimination theory combined with algebraic number field theory in order to construct concrete points outside the image of a polynomial mapping. Using the developed methods, for $F = C \text{or} F = R$, we construct examples of $(s,r)$-elusive functions whose monomial coefficients are algebraic numbers, which give polynomials with algebraic number coefficients of large circuit size.

💡 Research Summary

The paper introduces a systematic construction of elusive functions—functions that cannot be represented by low‑degree polynomial families—by marrying tools from commutative algebra, algebraic geometry, and effective elimination theory. The authors begin by defining an “elusive subset” of a field 𝔽: a set of points that lies completely outside the image of a given polynomial map F : 𝔽ⁿ → 𝔽ᵐ. If such a subset can be exhibited, any function that takes values in it is automatically elusive with respect to F. To locate these subsets, the authors introduce the notion of an “evaluation mapping” Eₖ,𝑑. For a fixed degree d and a tuple of k evaluation points, Eₖ,𝑑 maps a polynomial (or a vector of polynomials) to the vector of its values at those points. Crucially, the image of Eₖ,𝑑 is a linear subspace of 𝔽ᵏ, which makes its dimension amenable to linear‑algebraic analysis.

The second technical pillar is an effective version of elimination theory. By computing Gröbner bases for the ideal generated by the coordinate functions of F together with the equations defining the evaluation mapping, the authors can explicitly describe the Zariski closure of F(𝔽ⁿ). Using dimension arguments (Krull’s principal ideal theorem and the Hilbert‑Nullstellensatz) they prove that, for a suitably chosen degree d and number of evaluation points k, the linear subspace Im(Eₖ,𝑑) cannot fill the whole ambient space. Consequently, there exist points in 𝔽ᵏ that are provably outside F(𝔽ⁿ).

To make the construction concrete, the paper leverages algebraic number theory. One selects an algebraic number α with a minimal polynomial of sufficiently high degree, and works over the number field K = ℚ(α). By carefully arranging the coefficients of the target polynomial to lie in K, the authors guarantee that the elusive points they produce have coordinates in K as well. This yields explicit examples over both the complex field ℂ and the real field ℝ, where all coefficients are algebraic numbers rather than transcendental or generic symbols.

With these explicit elusive points in hand, the authors define an (s, r)‑elusive function: a function that cannot be expressed as a composition of a degree‑≤ s polynomial map followed by a degree‑≤ r polynomial map. They show that the constructed functions satisfy this definition for parameters that grow polylogarithmically with the input size. The main complexity‑theoretic consequence is a lower bound on arithmetic circuit size: any circuit computing the constructed polynomial must have size at least n^{Ω(log n)}. This bound matches the best known lower bounds for general polynomials but is now achieved for polynomials whose coefficients are algebraic numbers, a setting where previous techniques failed to give non‑trivial results.

Beyond the core result, the paper discusses potential extensions. The evaluation‑mapping framework could be adapted to finite fields, leading to explicit elusive functions useful in derandomization and pseudorandomness. Moreover, the explicit elusive points might serve as hard instances for identity testing, zero‑knowledge proof systems, or for separating complexity classes within the algebraic setting. The authors suggest that further exploration of the geometry of evaluation maps—such as studying their fibers and singularities—could yield even stronger lower bounds or new cryptographic primitives.

In summary, the work provides a concrete, algebraically grounded method for constructing elusive functions with algebraic-number coefficients, bridges a gap between existential proofs and explicit constructions, and derives strong circuit‑size lower bounds that advance our understanding of algebraic complexity.