Budan's theorem

In mathematics, Budan's theorem, named for François Budan de Boislaurent, is an early theorem for computing an upper bound on the number of real roots a polynomial has inside an open interval by counting the number of sign variations or sign changes in the sequences of coefficients.

Since 1836, the statement of Budan's theorem has been replaced in the literature by the statement of an equivalent theorem by Joseph Fourier, and the latter has been referred to under various names, including Budan's. Budan's original theorem forms the basis of the fastest known method for the isolation of the real roots of polynomials.

Budan's theorem is equivalent to Fourier's theorem. Though Budan's formulation preceded Fourier's, the name Fourier has usually been associated with it.

Given an equation in ${\displaystyle x}$, ${\displaystyle p(x)=0}$ of degree ${\displaystyle n>0}$, it is possible to make two substitutions ${\displaystyle x\leftarrow x+l}$ and ${\displaystyle x\leftarrow x+r}$ where ${\displaystyle l}$ and ${\displaystyle r}$ are real numbers so that ${\displaystyle l<r}$. If ${\displaystyle v_{l}}$ and ${\displaystyle v_{r}}$ are the sign variations in the sequences of the coefficients of ${\displaystyle p(x+l)}$ and ${\displaystyle p(x+r)}$ respectively then, provided ${\displaystyle p(r)\neq 0}$, the following applies:

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