Reflections on a Pair of Theorems by Budan and Fourier

Abstract

Isolation of the real roots of a polynomial equation is the process of finding real, disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. This process is quite important because, as J. B. J. Fourier pointed out, it constitutes the first step toward the solution of general equations of degree greater than four, the second step being the approximation of roots to any desired degree of accuracy. In the beginning of the 19th century F. D. Budan and J. B. J. Fourier presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval. Budan's theorem appeared in 1807 in the memoir Nouvelle methode pour la resolution des equations numeriques [10, p. 219], whereas Fourier's theorem was first published in 1820 in Le Bulletin des sciences par la Societe Philomatique de Paris, pp. 156, 181 [10, p. 223]. Due to the importance of these two theorems, there was a great controversy regarding priority rights. In his book (1859) Biographies of distinguished scientific men, p. 383, F. Arago informs us that Fourier deemed it necessary to have recourse to the certificates of early students of the Polytechnic School or Professors of the University in order to prove that he had taught his theorem in 1796, 1797 and 1803 [10]. Based on Fourier's proposition, C. Sturm presented in 1829 an improved theorem whose application yields the exact number of real roots which a polynomial equation without multiple zeros has within a real interval; thus he solved the real root isolation problem. Since 1830 Sturm's method has been the only one widely known and used, and consequently Budan's theorem was pushed into oblivion. To our knowledge, Budan's theorem can be found only in [16] and [61 whereas Fourier's proposition appears in almost all texts on the theory of equations. We feel that Budan's theorem merits special attention because it constitutes the basis of Vincent's forgotten theorem of 1836 which, in turn, is the foundation of our method for the isolation of the real roots of an equation [1], a method which far surpasses Sturm's in efficiency [2], [3]. In the discussion which follows we present separately, and without proofs, the classical theorems by Fourier and Budan and we indicate how they lead to the corresponding real root isolation methods. Some empirical results are also presented for comparison.