View source for Budan-Fourier theorem

The number of roots of an algebraic equation

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comprised in an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177402.png" />, is equal to or is smaller, by an even number, than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177403.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177404.png" /> is the number of changes in sign in the series of derivatives of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177405.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177406.png" />, i.e. in the series

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while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177408.png" /> is the number of changes in sign in this series at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017740/b0177409.png" />. Each multiple root is counted according to its multiplicity. Established by F. Budan (1822) and J. Fourier (1820).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Encyclopaedia of elementary mathematics'' , '''2. Algebra''' , Moscow-Leningrad  (1951)  pp. 331  (In Russian)</TD></TR></table>



====Comments====
An application of the Budan–Fourier theorem in numerical analysis may be found in [[#References|[a1]]], where it is used in the interpolation by spline functions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. de Boor,   I.J. Schoenberg,   "Cardinal interpolation and spline functions VIII. The Budan–Fourier theorem for splines and applications."  K. Bohmer (ed.)  G. Meinardus (ed.)  W. Schempp (ed.) , ''Spline functions'' , ''Lect. notes in math.'' , '''501''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.S. Householder,   "Unique triangularization of a nonsymmetric matrix"  ''J. Assoc. Comp. Mach.'' , '''5'''  (1958)  pp. 339–342</TD></TR></table>