Descartes' rule of signs

Descartes' rule of signs bounds a real polynomial’s positive roots, counted with multiplicity, by the number of sign changes in its nonzero coefficients. Applying it to the polynomial at −x gives a corresponding bound for negative roots.

Sign variation: The number of times consecutive entries in a sequence of nonzero real numbers have opposite signs. The rule uses sign variation in the coefficient sequence to bound positive roots.
Polynomial root-finding: The process of determining the roots of a polynomial exactly or approximately. The rule narrows possible positive and negative root counts before a full solution.
René Descartes: A French philosopher and mathematician whose work helped establish analytic geometry and modern algebraic methods. Descartes stated the rule in his 1637 work La Géométrie.
Budan's theorem: A theorem bounding the number of real roots in an interval by sign variations in a sequence of polynomial derivatives. It extends sign-variation reasoning to intervals rather than only positive and negative roots.
Polynomial root: A number at which a polynomial evaluates to zero. The rule bounds how many polynomial roots lie on each side of zero.
Sturm's theorem: A theorem that counts distinct real roots of a polynomial in an interval using a sequence of polynomials. It can determine exact interval root counts when Descartes' rule gives only an upper bound.
La Géométrie: René Descartes' 1637 mathematical work introducing coordinate methods linking algebra and geometry. This work presented the sign rule as part of a method for understanding polynomial roots.
Vincent's theorem: A root-isolation result that uses fractional linear transformations and sign variations to isolate positive roots. Repeated transformations can sharpen sign-based bounds into certified root isolation.
Multiplicity of a root: The number of times a factor corresponding to a root divides a polynomial. Roots are counted with multiplicity when comparing the bound with actual positive roots.
Bisection method: A numerical method that repeatedly halves an interval whose endpoint function values have opposite signs. A sign change establishes an interval containing a root, which can then be approximated by bisection.

Linked from 5 pages