fix(polys): fix multi factorisation over QQ<a>

[oscarbenjamin]

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Fixes gh-26497

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Fixes gh-26497

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[oscarbenjamin]

The two doctests that are failing are fine. They can just be updated. The output of dmp_sqf_norm is not uniquely defined so it is fine to return different outputs as long as they satisfy the invariant.

See the Trager 1976 reference noted in the docstring for explanation of what this function is doing.

[oscarbenjamin]

Should probably check dmp_degree_in > 1 first

[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

| Change   | Before [2487dbb5]    | After [bd0b3e6f]    |   Ratio | Benchmark (Parameter)                                                |
|----------|----------------------|---------------------|---------|----------------------------------------------------------------------|
| -        | 67.4±0.8ms           | 43.6±0.1ms          |    0.65 | integrate.TimeIntegrationRisch02.time_doit(10)                       |
| -        | 66.8±1ms             | 42.9±0.4ms          |    0.64 | integrate.TimeIntegrationRisch02.time_doit_risch(10)                 |
| +        | 19.2±0.5μs           | 29.9±0.4μs          |    1.56 | integrate.TimeIntegrationRisch03.time_doit(1)                        |
| -        | 5.32±0.01ms          | 2.82±0.01ms         |    0.53 | logic.LogicSuite.time_load_file                                      |
| -        | 73.6±0.4ms           | 28.9±0.3ms          |    0.39 | polys.TimeGCD_GaussInt.time_op(1, 'dense')                           |
| -        | 25.7±0.08ms          | 16.9±0.08ms         |    0.66 | polys.TimeGCD_GaussInt.time_op(1, 'expr')                            |
| -        | 74.5±0.5ms           | 28.7±0.3ms          |    0.38 | polys.TimeGCD_GaussInt.time_op(1, 'sparse')                          |
| -        | 255±1ms              | 124±0.6ms           |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'dense')                           |
| -        | 257±1ms              | 126±0.4ms           |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'sparse')                          |
| -        | 652±9ms              | 373±1ms             |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'dense')                           |
| -        | 653±4ms              | 373±1ms             |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'sparse')                          |
| -        | 487±2μs              | 290±1μs             |    0.6  | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(1, 'dense')            |
| -        | 1.77±0.01ms          | 1.06±0ms            |    0.6  | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(2, 'dense')            |
| -        | 5.80±0.06ms          | 3.06±0.01ms         |    0.53 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(3, 'dense')            |
| -        | 445±2μs              | 233±2μs             |    0.52 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(1, 'dense')               |
| -        | 1.49±0.01ms          | 674±6μs             |    0.45 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(2, 'dense')               |
| -        | 4.83±0.05ms          | 1.65±0.01ms         |    0.34 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(3, 'dense')               |
| -        | 376±2μs              | 208±1μs             |    0.55 | polys.TimeGCD_SparseGCDHighDegree.time_op(1, 'dense')                |
| -        | 2.43±0.01ms          | 1.22±0.01ms         |    0.5  | polys.TimeGCD_SparseGCDHighDegree.time_op(3, 'dense')                |
| -        | 9.96±0.08ms          | 4.28±0.02ms         |    0.43 | polys.TimeGCD_SparseGCDHighDegree.time_op(5, 'dense')                |
| -        | 358±3μs              | 169±1μs             |    0.47 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(1, 'dense')            |
| -        | 2.50±0.01ms          | 924±6μs             |    0.37 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(3, 'dense')            |
| -        | 9.43±0.08ms          | 2.62±0.01ms         |    0.28 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(5, 'dense')            |
| -        | 1.02±0.01ms          | 424±3μs             |    0.42 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'dense')           |
| -        | 1.71±0.01ms          | 516±3μs             |    0.3  | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'sparse')          |
| -        | 5.91±0.1ms           | 1.79±0.01ms         |    0.3  | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'dense')           |
| -        | 8.61±0.07ms          | 1.70±0.01ms         |    0.2  | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'sparse')          |
| -        | 290±1μs              | 66.3±1μs            |    0.23 | polys.TimePREM_QuadraticNonMonicGCD.time_op(1, 'sparse')             |
| -        | 3.42±0.04ms          | 393±3μs             |    0.11 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'dense')              |
| -        | 3.96±0.02ms          | 283±0.7μs           |    0.07 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'sparse')             |
| -        | 7.02±0.05ms          | 1.26±0.01ms         |    0.18 | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'dense')              |
| -        | 8.70±0.06ms          | 837±2μs             |    0.1  | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'sparse')             |
| -        | 5.04±0.04ms          | 3.00±0.03ms         |    0.6  | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(2, 'sparse') |
| -        | 12.0±0.04ms          | 6.63±0.07ms         |    0.55 | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'dense')  |
| -        | 22.4±0.06ms          | 9.01±0.02ms         |    0.4  | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'sparse') |
| -        | 5.22±0.02ms          | 865±3μs             |    0.17 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(1, 'sparse')    |
| -        | 12.6±0.04ms          | 7.02±0.02ms         |    0.56 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(2, 'sparse')    |
| -        | 101±0.5ms            | 26.0±0.09ms         |    0.26 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'dense')     |
| -        | 167±0.6ms            | 53.8±0.8ms          |    0.32 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'sparse')    |
| -        | 175±1μs              | 113±0.5μs           |    0.65 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'dense')      |
| -        | 363±4μs              | 215±1μs             |    0.59 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'sparse')     |
| -        | 4.33±0.02ms          | 838±6μs             |    0.19 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'dense')      |
| -        | 5.28±0.05ms          | 380±1μs             |    0.07 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'sparse')     |
| -        | 19.5±0.1ms           | 2.79±0.01ms         |    0.14 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'dense')      |
| -        | 22.5±0.09ms          | 622±4μs             |    0.03 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'sparse')     |
| -        | 483±5μs              | 138±0.7μs           |    0.29 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(1, 'sparse') |
| -        | 4.65±0.02ms          | 616±1μs             |    0.13 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'dense')  |
| -        | 5.31±0.06ms          | 140±0.6μs           |    0.03 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'sparse') |
| -        | 13.2±0.09ms          | 1.30±0.01ms         |    0.1  | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'dense')  |
| -        | 14.2±0.06ms          | 144±1μs             |    0.01 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'sparse') |
| -        | 134±2μs              | 74.3±0.3μs          |    0.55 | solve.TimeMatrixOperations.time_rref(3, 0)                           |
| -        | 253±2μs              | 87.9±0.4μs          |    0.35 | solve.TimeMatrixOperations.time_rref(4, 0)                           |
| -        | 24.0±0.2ms           | 10.2±0.04ms         |    0.43 | solve.TimeSolveLinSys189x49.time_solve_lin_sys                       |
| -        | 28.2±0.1ms           | 15.3±0.06ms         |    0.54 | solve.TimeSparseSystem.time_linsolve_Aaug(20)                        |
| -        | 54.7±0.3ms           | 24.7±0.2ms          |    0.45 | solve.TimeSparseSystem.time_linsolve_Aaug(30)                        |
| -        | 28.1±0.1ms           | 15.2±0.1ms          |    0.54 | solve.TimeSparseSystem.time_linsolve_Ab(20)                          |
| -        | 54.5±0.2ms           | 24.5±0.1ms          |    0.45 | solve.TimeSparseSystem.time_linsolve_Ab(30)                          |

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).