Verified Note

Divisibility of Separated Polynomials

A criterion for when f(X)-g(Y) divides p(X)-q(Y), expressed through descent to a common one-variable outer polynomial.

Summary

Result and mechanism

The note turns a two-variable divisibility condition into a one-variable descent statement.

Main result

Under the stated hypotheses, f(X)-g(Y) divides p(X)-q(Y) if and only if p and q are obtained by composing f and g with a common outer polynomial.

Contribution type
New argument; Lean formalization
Contribution
Common-outer-polynomial divisibility criterion for separated polynomial differences, proved through free-module residue expansion and diagonal reduction.
Mechanism
free-module residue rectangle and diagonal reduction
Lean theorem
SeparatedPolynomialDifferences.sep_dvd_iff_exists_common_outer
Lean and mathlib
Lean 4.30.0-rc2, mathlib v4.30.0-rc2
Build command
lake build