Budan's theorem

Budan's theorem bounds the number of distinct real roots of a polynomial in an interval by the decrease in sign variations among its successive derivatives at the endpoints.

Sign variation: The number of sign changes between consecutive nonzero terms in an ordered sequence. Budan's bound compares sign variations in derivative-value sequences at the two endpoints.
Polynomial: An expression formed from constants and variables using addition, subtraction, and nonnegative integer powers. Budan's theorem applies to polynomial functions and their derivative sequences.
Real-root isolation: The process of finding disjoint intervals that each contain exactly one real root of a polynomial. Budan bounds help reject intervals or guide subdivision during root isolation.
François Budan de Boislaurent: A French mathematician whose work on polynomial equations gave rise to Budan's theorem. His name is attached to the endpoint derivative sign-variation bound.
Descartes' rule of signs: A rule that bounds positive roots of a polynomial by sign changes in its coefficient sequence. It supplies the sign-counting principle that Budan's theorem extends from coefficients to endpoint derivatives.
Real root: A real number at which a given polynomial evaluates to zero. The theorem bounds how many such roots lie within the chosen interval.
Bisection method: A numerical method that repeatedly halves an interval to locate a zero of a continuous function. A Budan bound can assess whether subdivided intervals may contain polynomial roots.
Joseph Fourier: A French mathematician and physicist whose work included a theorem on real-root bounds for polynomials. Fourier independently developed the derivative-based result often called the Budan–Fourier theorem.
Polynomial derivative: The derivative of a polynomial, formed by differentiating each term. Successive derivatives provide the endpoint values whose signs determine the bound.
Multiplicity of a root: The number of times a polynomial's linear factor occurs in its factorization at a root. Budan's variation difference bounds roots counted with multiplicity, hence also distinct roots.

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