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In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting zero coefficients), and the difference ...
Descartes' rule of signs asserts that the difference between the number of sign variations in the sequence of the coefficients of a polynomial and the number of its positive real roots is a nonnegative even integer. It results that if this number of sign variations is zero, then the polynomial does not have any positive real roots, and, if this ...
For 0 < k < n, Descartes' rule of signs implies that () either has two positive real roots that are not multiple, or is nonnegative for every positive value of x. So, the above result may be applied only in the first case. If , <, are these two roots, the above result implies that
All results described in this article are based on Descartes' rule of signs. If p(x) is a univariate polynomial with real coefficients, let us denote by # + (p) the number of its positive real roots, counted with their multiplicity, [1] and by v(p) the number of sign variations in the sequence of its coefficients. Descartes's rule of signs ...
[4]: 23–24 The specific topics treated bear witness to the special interests of Pólya (Descartes' rule of signs, Pólya's enumeration theorem), Szegö (polynomials, trigonometric polynomials, and his own work in orthogonal polynomials) and sometimes both (the zeros of polynomials and analytic functions, complex analysis in general).
Sturm's theorem provides a way for isolating real roots that is less efficient (for polynomials with integer coefficients) than other methods involving Descartes' rule of signs. However, it remains useful in some circumstances, mainly for theoretical purposes, for example for algorithms of real algebraic geometry that involve infinitesimals. [3]
Taken together with Descartes' Rule of Signs, this leads to an upper bound on the number of the real roots a polynomial has inside an open interval. Although Budan's Theorem , as this result was known, was taken up by, among others, Pierre Louis Marie Bourdon (1779-1854), in his celebrated algebra textbook, it tended to be eclipsed by an ...
2 Moved back to "Descartes' rule of signs" ... 3 Second Example. 1 comment. 4 Comment. 2 comments. 5 Complex Roots. 6 comments. 6 Original source + unchanging signs ...