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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. ==Sign variation== :Let ''c''0, ''c''1, ''c''2, ... be a finite or infinite sequence of real numbers. Suppose ''l'' < ''r'' and the following conditions hold: # If ''r'' = ''l''+1 the numbers ''cl'' and ''cr'' have opposite signs. # If ''r'' ≥ ''l''+2 the numbers ''cl+1'', ..., ''cr−1'' are all zero and the numbers ''cl'' and ''cr'' have opposite signs. : This is called a ''sign variation'' or ''sign change'' between the numbers ''cl'' and ''cr''. : When dealing with the polynomial ''p''(''x'') in one variable, one defines the number of sign variations of ''p''(''x'') as the number of sign variations in the sequence of its coefficients. Two versions of this theorem are presented: the ''continued fractions'' version due to Vincent, and the ''bisection'' version due to Alesina and Galuzzi. This statement of the ''continued fractions'' version can be found also in the Wikipedia article Budan's theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Vincent's theorem」の詳細全文を読む スポンサード リンク
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