Vincent's theorem

In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients.

Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them.

Two versions of this theorem are presented: the continued fractions version due to Vincent, and the bisection version due to Alesina and Galuzzi.

If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form

where ${\displaystyle a_{1},a_{2},a_{3},\ldots }$ are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:

Moreover, if infinitely many numbers ${\displaystyle a_{1},a_{2},a_{3},\ldots }$ satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.

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