Generator

eliminate

A generator in the algebra library, called 5 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

eliminate is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Two coefficient arrays, one with determinant zero and one without. The Sylvester matrices of two pairs of polynomials drawn as grids of coefficients, one pair sharing a root and one not, with each determinant computed in whole numbers and checked against whether a shared root exists.

`show: "curves"`

Two curves, and the single-variable equation their crossings satisfy. Two plane curves drawn crossing, with the polynomial obtained by eliminating one variable plotted below on the same horizontal axis, so its roots sit under the crossings.

`show: "discriminant"`

Three cubics, and the determinant that finds the repeated root. Three cubic curves drawn against a horizontal axis, one of them touching it rather than crossing, with the determinant of each polynomial and its derivative written underneath.

`show: "sweep"`

The determinant of x² − 2 against x² + c, swept. The resultant of a fixed quadratic and a sliding one plotted against the sliding one's constant term, with the places it crosses zero marked and each one confirmed to be a shared root.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

a shared root is found exactly when the determinant says there is one ×1
all 2 roots of the eliminated polynomial are accounted for ×1
both polynomials have a degree of at least one ×1
each step of the division lowers the degree ×1
every root of the eliminated polynomial is the abscissa of a real crossing ×1
every zero of the swept determinant is a parameter at which a root is shared ×1
the determinant equals the second polynomial evaluated at every root of the first ×1
the determinant vanishes exactly when the two share a factor ×1
the eliminated polynomial is the resultant at every sampled x ×1
the fixed polynomial is an integer quadratic ×1
the pair of curves is one this family knows ×1
the resultant of a polynomial and its slope vanishes exactly at a repeated root ×1
the sweep runs upward over a range of 1 to 20 ×1
the view is one the family draws ×1
two pairs of integer polynomials, each of degree 1 to 3 ×1
two to four integer cubics ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Algebra

A shared root, found without finding it

Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.

The whole library · What the figures prove