Factor a polynomial in any number of variables (#746 item 43)

[Rafael-SOWNet]

#1055 factored a polynomial in two variables by Kronecker's substitution, because two is the case #746 item 43 names. Nothing in the method is about two.

The exponent pair (i, j) was packed as i + s·j — a two-digit numeral in radix s. An exponent vector of any length is a numeral in mixed radix: radices d_i + 1, place values s_0 = 1, s_(i+1) = s_i · (d_i + 1). A factor has degree at most d_i in each variable because it divides the polynomial, so the map stays injective on every monomial that can appear in the polynomial or in any of its factors, whatever the number of variables.

So the encoding and decoding become loops over the variables instead of two statements, the class is KroneckerFactorization rather than BivariateFactorization, and the caller stops refusing a third variable.

What it answers now

	before	now
Factor("x ^ 2 - (y + z) ^ 2", "x")	null	(x + y + z) * (x - y - z)
Factor("x ^ 2 + 2 * x * y + y ^ 2 - z ^ 2", "x")	null	(x + y + z) * (x + y - z)
Factor("x * y - x - y + 1", "x")	null	(y - 1) * (x - 1)
Factor("(x + y) * (x + z + w)", "x")	null	(w + x + z) * (x + y)
Factor("(x + y) * (x + z) * (x + w)", "x")	null	(x + y) * (w + x) * (x + z)

Everything two-variable answers exactly as before.

The ceiling moves sharply, and it is still a refusal

The image has degree Π (d_i + 1) - 1 — a product, not a sum — against the one-variable factoriser's MaxDegree of 32. Three variables of degree 2 fit (27); four do not (81). A new test pins three shapes that factor mathematically and are declined:

(x + y + z + w) * (x - y)              degrees (2,2,1,1) -> image degree 35
(x + y) * (x + z) * (x + w) * (x + v)  degrees (4,1,1,1,1) -> image degree 79
x ^ 12 - y ^ 12                        two variables, past it on its own

A wrong answer is still not among the things this can do. Every candidate is checked by exact division before it is kept, and the assembled factors are divided back into the input. Lifting the ceiling is Hensel lifting with an evaluation homomorphism, which is a different piece of work and stays open on item 43.

Also

BREAKING-CHANGES.md held two entries disagreeing about x ^ 2 - y ^ 2 — #1053's said it is null and needs factorisation over ℚ(y), #1055's said it is (x + y) * (x - y). The second had overtaken the first without the first being updated. The earlier entry now points at the later one instead of contradicting it.

Full suite: Failed: 0, Passed: 8583, Skipped: 14, Total: 8597.

🤖 Generated with Claude Code

https://claude.ai/code/session_01Bjumi5K7fg8yx6UK1mZTQd