Explicit rigid matrices over the rationals

Adapt the deterministic polynomial-time algorithm for the Remote Point Problem over the rational numbers to construct explicit rigid matrices over Q whose integer or rational entries have polynomial bit complexity.

Background

The paper notes that constructing rigid matrices over Q remains an important problem. Semi-explicit rigid matrices over Q are known when the entries have doubly exponential magnitude and therefore exponential bit complexity, but the paper does not obtain matrices with polynomial-bit-complexity entries.

The proposed direction is to determine whether the techniques underlying the optimal rational remote-point algorithm can yield a fully explicit construction with substantially smaller representation complexity.

References

Can the algorithm from \cref{thm:rpp-over-Q} be adapted to obtain an explicit construction of rigid matrices over $Q$? It is possible to obtain ``semi-explicit'' rigid matrices over $Q$ in which the entries are integers of doubly-exponential magnitude (and hence exponential bit complexity) . The techniques for proving that such matrices are rigid are somewhat reminiscent of our proof of \cref{thm:rpp-over-Q}. Nevertheless, we are not able to obtain matrices whose entries are integers (or rationals) with polynomial bit complexity.

— Improved Algorithms for the Remote Point Problem  (2609.25765 - Volk, 22 Sep 2026) in Section Open Problems, second item