Minor — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Also named here as sylvesters identity — the same set of essays touches all of them, so they are one junction rather than several.
Every intermediate is a minor
Fraction-free elimination divides by the previous pivot at every step and the division is always exact. Not usually, not for these entries — always, because the number being divided is a determinant with that pivot as a factor, which is a theorem and is checked here against the minors themselves.
A bound on every intermediate at once
Fraction-free elimination's intermediates are minors of the original, which is a theorem about exactness. It is also a bound: Hadamard's inequality applies to every minor, so one inequality bounds the whole run before it starts. The bound on the k-th step is the one on (k+1)×(k+1) minors, not the one on the whole matrix — and on a 10×10 with entries in ±6 the difference is ten bits, with the run reaching 2.7 bits a step against the bound's 3.4.
Three orders and one last entry
Over the integers there is no stability to pivot for, so a fraction-free elimination swaps rows only when the pivot is zero. Choosing a pivot for length instead does change the sequence of minors — the smallest-nonzero rule makes seven exchanges where the natural order makes none and keeps the profile two bits lower through the middle. It cannot change the peak. The last entry of the elimination is the determinant, and the determinant does not know what order it was computed in.
Named alongside it
The objects these essays reach for when they reach for this one.
Bit lengthDeterminantExact arithmeticFraction-free eliminationHadamard boundSylvesters identityExact ground truthFlop countGaussian eliminationPartial pivotingPermutation