A linear system with symbols in its entries is solved over polynomials in the symbols, fraction-free, so that its zeros are zeros and its answers are one fraction each

[Rafael-SOWNet]

The elimination behind partial fractions, the reciprocal substitution and
the cube-root ansatz runs on entities, whose arithmetic collects nothing:
with symbols in the entries its zero test is numeric, so it pivots on what
may be zero and declines a consistent system it has left 2b - 2b in, and
its answers are what the arithmetic wrote. 1/((x + 1) sqrt(x^2 + x + b))
through the Euler substitution came back correct in 24 KB, with
partial-fraction coefficients like (2b - 2b) beside terms that were zero;
1/((x + a) sqrt(x^2 + b x + c)) in 77 KB and thirteen seconds.
1/((x^2 + a)^2 (x + 1)) was declined, and a test pinned that.

Where every entry is a polynomial over Q in the symbols, the system is now
solved as such first: Bareiss' fraction-free elimination keeps every
intermediate a polynomial -- a minor of the augmented matrix, divided
exactly by the previous pivot -- so every zero is a zero, and the
Gauss-Jordan form of it (Nakos, Turner and Williams, 1997) reduces above
the pivot too, so at the end each pivot row reads d x_k = n_k with d the
last pivot: one fraction per unknown, both sides primitive with the
rational they were scaled by in front, in lowest terms where the gcd
finds them, and no back-substitution to compound them. An entry outside
that ring -- a number outside Q, a symbol under a root or below the bar --
or an intermediate past the term ceiling hands the system back to the
elimination on entities, unchanged.

1/((x + a) sqrt(x^2 + b x + c)) is one arctangent or one logarithm per
arm of a piecewise on the discriminant, under 2 KB, in half a second;
1/((x^2 + a)^2 (x + 1)) is answered, on both signs of a. The test that
pinned its decline is repinned to the answer.

Rubi sample (463 problems, 5 s budget), against the #1312 master this branch is cut from:

master at #1312     404/463, 0 wrong, 0 timeouts, 99-100 s
with this           405/463, 0 wrong, 0 timeouts, 97 s

One more, nothing lost: Moses 114, (-A^2 - B^2)/(B (1 + w^2)^2 (1 - (A^2 + B^2) w^2/(B^2 (1 + w^2)))), 3.5 s to decline before and 0.3 s to answer now. No other problem moved by more than noise.

Independent of #1313, which pins the symbols and solves on the support it finds; the two compose -- the pinned solve calls this one -- and conflict only in the file, on the line both add before the elimination.

The five test suites are green.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura