Polynomial roots — the series

The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

A loop that cannot miss the middle

Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.

part 1 · algebra

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.

part 2 · algebra

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.

part 3 · algebra

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.

part 4 · algebra

The roots of the derivative inside the hull of the roots. Three polynomials of degree five, each drawn as its roots with their convex hull shaded, and the four roots of its derivative marked. Every root of the derivative lies inside the hull.

The roots of the slope stay inside

Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.

part 5 · algebra

The roots of random polynomials of degree 30 and 100: few are real. degree 30: 2 real roots at -2.335, -1.074; degree 100: 2 real roots at -0.327, 0.994.

Almost none of the roots are real

Pick the coefficients of a polynomial of degree a thousand at random, each one an independent draw from the bell curve, and ask how many of its thousand roots are real. The answer is about five. Mark Kac found in 1943 that the average grows only like (2/π) ln n, and the reason can be read off the real line itself: the real roots crowd towards +1 and −1 and spread evenly on a logarithmic scale of distance from them.

part 6 · algebra

The roots of every polynomial of degree 11 with coefficients ±1. 22528 roots; radii from 0.500 to 2.000; 42% within 0.1 of the unit circle.

Random roots crowd onto the circle

The roots of a random polynomial are not scattered across the plane. They gather within about 1/n of the unit circle, and their directions spread evenly round it. Paul Erdős and Pál Turán proved in 1950 that this is not a fact about randomness at all: any polynomial whose coefficients are all of roughly one size has roots whose directions are nearly even, and the inequality says exactly how nearly, in terms of nothing but the coefficients.

part 7 · algebra

Three polynomials of signs around the circle. The size of the polynomial around the unit circle divided by √64 for all-plus, random and Rudin–Shapiro sign polynomials of length 64; maxima 8.00, 2.50, 1.41.

The flattest polynomials of signs

A polynomial whose coefficients are all +1 or −1 has average size √n on the unit circle. Keeping it near √n everywhere is the problem Littlewood posed: the Rudin–Shapiro polynomials never exceed √2 times it, searching every sign pattern up to length 22 finds the best are the Barker sequences, and whether the maximum can come arbitrarily close to √n is still open.

part 8 · algebra