Gaussian elimination — the algorithm that solves systems of linear equations — occasionally amplifies small entries into large ones during the pivoting process. The growth factor measures the worst amplification: the largest entry that appears during elimination divided by the largest entry in the original matrix. For a 5×5 matrix with complete pivoting, numerical optimization by Day and Peterson (1988) and Gould (1991) found the maximum growth factor: 4.1325...
The number was known to five decimal places. Its exact nature was unknown. Was it transcendental? Algebraic? The product of some simple combination of known constants?
Lanza, Rump, and Uherka (arXiv:2602.20390) answer: it is the root of a specific 61st degree polynomial. They find the polynomial exactly using a hybrid of numerical optimization (JuMP), symbolic algebra (Gröbner bases), and rigorous verification (interval arithmetic). The number that was computed numerically 37 years ago turns out to be algebraic, and the polynomial that defines it has degree 61.
The degree is not arbitrary — it reflects the combinatorial complexity of the constraint structure. The growth factor optimization for a 5×5 matrix has equality constraints that define an algebraic variety. The polynomial is the minimal polynomial of the optimal point on that variety. The number 61 is a property of the problem's geometry, not an approximation artifact.
The upper bound on the growth factor for 5×5 matrices is also improved — from 4.94 to 4.84 — though the combinatorial explosion prevents proving this is the global maximum.
The general observation: a number found by numerical optimization may be algebraically exact — the root of a polynomial whose degree reflects the combinatorial complexity of the optimization landscape. The number appears transcendental only because the polynomial that defines it is too complicated to find without hybrid computational methods. Exact structure can hide behind numerical approximation for decades.