Matpic
mathematics, in pictures
Concept

Symmetric function

A polynomial in several variables that is unchanged when the variables are permuted among themselves. The sum and the product of a polynomial's roots are the basic examples, and every symmetric function of the roots is a polynomial in the coefficients.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

algebra · Polynomial roots 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.

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.

algebra · Polynomial roots A year of birthdays with a seasonal swing of ±50%, peaking in September. Bars for the 365 days of a model calendar with a seasonal swing of ±50%, peaking in September, drawn as each day's excess or shortfall against an even year. A shared birthday among 23 people has chance 54.86% against 50.73% for the even year, the first group with an even chance is 22, and the calendar behaves like 324.4 equally likely days.

Any unevenness brings the match sooner

Real birthdays are not spread evenly across the year, and every such departure pushes the famous twenty-three down rather than up. The proof is one move on two days at a time, and what it leaves behind is a single number — the one ecologists use to count species.

probability · Birthday problem Every quadratic is a point. The plane of monic quadratics x² + px + q with p across and q up. The parabola q = p²/4 divides it: the region below, shaded, holds the equations with two real roots, the curve itself the ones with a repeated root, and the region above the ones with none. 5 equations are marked and labelled.

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

algebra · Completing the square The powers of 4 roots, added: 1, −1, 4, −5. For k from 1 to 4, the k-th powers of the roots of a degree-4 polynomial drawn as arrows placed tip to tail. Each walk ends on the real axis at a whole number, the k-th power sum, which the coefficients determine.

Every power sum, from the coefficients alone

Raise the roots of a polynomial to the k-th power and add them. However the roots turn, the total is a whole number when the coefficients are, and Newton's identities produce it from the coefficients one step at a time — no root is ever found. Run the rule on x³ − x − 1 and out comes Perrin's sequence, whose terms know which numbers are prime, nearly.

algebra · Polynomial roots The three pairings of the roots of x⁴ + x + 1. Three panels each showing the same four roots of a quartic in the complex plane, joined in a different way into two pairs, with the value of the sum of the pair products beneath.

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

algebra · Galois correspondence The sine rebuilt from its zeros. The curve sin πx and three partial products of Euler's formula, with 1, 3 and 10 pairs of factors; each matches the sine between its zeros and diverges outside them.

The sine, rebuilt from its zeros

A polynomial is a product of factors, one for each root. Euler treated the sine the same way — one factor for each place the wave crosses the axis — and the product he wrote down is correct, though the zeros alone do not justify it. Multiplied out, it hands over the sum of the reciprocal squares, π²/6, and every even power after it.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

PolynomialDiscriminantRootsCoefficientComplex numbersBirthday problemCollisionComplementary countingCompleting the squareConvergenceConvexityCubic

All concepts