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ModularElimination


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Description

Summary

TrySolveOverPolynomials(AngouriMath.Entity[][],AngouriMath.Entity[],AngouriMath.Entity[]@)'s reading and elimination run first on the
image of the system modulo LiftingPrime, to learn whether the ones over
Q decline: the same readings, pivots, products and exact divisions, in
residues that cost a machine multiplication each where the rationals of a minor cost
a greatest common divisor. The Bareiss intermediates are minors, so each is the image
of the one over Q and has no monomial that one lacks; a product of more than
MaxIntermediateTerms terms here, a quotient of
more, an exponent past MaxDegree, an entry the
reading refuses or a row that reduces to 0 = c is therefore one there too, and
the solve over Q refuses it. Anything else -- an image that does not exist, a
division that does not come out, a remainder too large to hold -- says nothing, and
the solve over Q runs as it did.

Remarks

The images follow the elimination over Q as long as no nonzero polynomial of
it vanishes modulo the prime, every one of its coefficients a multiple of
2^31 - 1; where one did, a pivot could differ and a system be declined that
would have been solved, and nothing is ever answered on the strength of the images.
The Hermite reductions of Rubi's 1/(a + b cosh(x) + c sinh(x))^n under
u = e^x are systems of twenty rows in three symbols whose minors pass four
thousand terms by the sixth pivot: the elimination over Q spent seconds on
each before declining it, eight of them for n = 3, and the rule the
integrator reaches after the decline answers in milliseconds.
https://github.com/asc-community/AngouriMath/issues/233

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