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Summary: Bull. Math. Soc. Sci. Math. Roumanie
Tome 53(101) No. 3, 2010, 201211
Counting the number of real roots in an interval with
Vincent's theorem
by
Alkiviadis G. Akritas and Panagiotis S. Vigklas
Dedicated to the memory of Laurent¸iu Panaitopol (1940-2008)
on the occasion of his 70th anniversary
Abstract
It is well known that, in 1829, the French mathematician Jacques Charles
Fran¸cois Sturm (1803-1855) solved the problem of finding the number of
real roots of a polynomial equation f(x) = 0, with rational coefficients and
without multiple roots, over a given interval, say ]a, b[. As a byproduct,
he also solved the related problem of isolating the real roots of f(x). In
1835 Sturm published another theorem for counting the number of complex
roots of f(x); this theorem applies only to complete Sturm sequences and
was recently extended to Sturm sequences with at least one missing term.
Less known, however, is the fact that Sturm's fellow countryman and
contemporary Alexandre Joseph Hidulphe Vincent (1797-1868) also pre-
sented, in 1836, another theorem for the isolation (only) of the positive
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