In mathematics, Sturm's theorem is a symbolic procedure to determine the number of unique real roots of a polynomial.

It was named for its discoverer, Jacques Charles François Sturm.

Whereas the fundamental theorem of algebra readily yields the number of real or complex roots of a polynomial, counted according to their multiplicities, Sturm's theorem deals with real roots and disregards their multiplicities.

To apply Sturm's theorem, you first construct a Sturm chain from a polynomial

.

A Sturm chain is the sequence of intermediary results when applying Euclid's algorithm to X and its derivative X₁ = X'.

To obtain the Sturm chain, you compute

i.e., you successively take the remainders with polynomial division and change theirs signs. Each X_i is a polynomial of degree at least one, hence the algorithm terminates. X_r then is the GCD of X and its derivative. If X only had simple roots, it will be a constant. The Sturm chain then is

.

Let w(ξ) be the number of sign changes (zeroes are not counted) in the sequence

.

Sturm's theorem then states that for two real numbers

a < b,

not roots of X, the number of roots in the interval

[a,b]

is

w(b) - w(a).

This can be used to compute the total number of real roots a polynomial has (to use, for example, as an input to a numerical root finder) by choosing a and b appropriately — for example, all real root of a polynomial with coefficients a_i are in the interval [ - M,M], where

.

Alternatively, you can use the fact that for large x, the sign of a polynomial

is <

math>\sgn(a_n)</math>,

whereas

sgn(P( - x)) = sgn(( - 1)ⁿa_n).

In this way, simply counting the sign changes in the leading coefficients in the Sturm chain readily gives the number of distinct real roots of a polynomial.