29.6. Performance🔗

Fixed-field arithmetic is polynomial arithmetic and scales as the degree suggests: multiplication is quadratic in the degree and inversion cubic with a logarithmic factor for coefficient growth. Everything that isolates roots is dominated by the isolation, so those rows are fixed inputs rather than asymptotics.

operation	input	time
`PolyQuot` multiplication	degree 16, dense coordinates	0.12 ms
`PolyQuot` inversion	degree 16, dense coordinates	0.71 ms
`PolyQuot.approx`	degree 128, fixed precision	17 ms
`ZPoly.algebraicRoots`	`X⁴ − 10X² + 1`, four roots	44 ms
`AlgebraicRoot.exact`	a root of `∏ (X² − p)`, `p ≤ 13`, degree 12	1.9 ms
`AlgebraicRoot.exact`	a root of `X⁸ − 2` inside `(X⁸ − 2)(X + 3)`	310 ms
`AlgebraicRoot.add`	a root of `X⁶ − 2` plus `√3`, degree product 12	4.5 s
`AlgebraicPoly.roots`	dense degree 6, one `√2` coefficient	6.0 s

Medians on chungus2 from the exports recorded in reports/hex-number-field-performance.md in the hex-dev repository; regenerate with .lake/build/bin/hexnumberfield_bench run Hex.NumberFieldBench.<target>. The last three rows spend over ninety percent of their time in root isolation of the resultant or eliminant, and the X⁸ − 2 exactification spends its time re-isolating candidates rather than factoring.