Real roots, printed decimals and algebraic degree of the Chen–Edelman–Urschel growth-factor polynomials

Abstract

Chen, Edelman and Urschel have recently attached to the maximal growth factor of Gaussian elimination with complete pivoting on 5 × 5 matrices an explicit integer polynomial P₅ of degree 61, conjecturing that the growth factor is its unique real root in [4,5]; their method also produces a sextic P₇ at n=7. We determine the complete real-root structure of both polynomials: P₅ has exactly seven real roots and P₇ exactly four, each isolated inside an interval with integer endpoints by a one-sign-change certificate. We show that P₇ is irreducible over ℚ, so that the value their method produces at n=7 is an algebraic number of degree exactly six with minimal polynomial P₇/3; the source exhibits the sextic but leaves minimality open, and the general bound it proves for the algebraic degree of the n × n growth factor reads 14⁴⁸ at n=7. We also record two errata to the source, neither of which affects any of its conclusions: the value printed "to 200 decimal places" for the n=5 root is correct to 182 places and exceeds the root by between 2 · 10⁻¹⁸³ and 3 · 10⁻¹⁸³ (the 78 decimals printed at n=7 are, by contrast, all correct), and the index at which the sign block of the source's own Descartes computation turns over is 9, not 10. Every theorem and proposition below is checked by the Lean 4 kernel on exact integer arithmetic; no floating point and no compiled evaluation enters any proof.

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