Historical account and ultra-simple proofs of Descartes's rule of signs, De Gua, Fourier, and Budan's rule
It may seem a funny notion to write about theorems as old and rehashed as Descartes's rule of signs, De Gua's rule or Budan's. Admittedly, these theorems were proved numerous times over the centuries. However, despite the popularity of these results, it seems that no thorough and up-to-date historical account of their proofs has ever been given, nor has an effort been made to reformulate the oldest demonstrations in modern terms. The motivation of this paper is to put these strongly related theorems back in their historical perspective. More importantly, we suggest a way to understand Descartes's original statement, which yet remains somewhat of an enigma. We found that this question is related to a certain way of counting the alternations and permanences of signs of the polynomial coefficients, and may have been the convention used by Descartes. Remarkably, this convention not only provides a ultra-simple proof of Descartes's rule, but it can also be used to simplify the proofs of the titular theorems. Without claiming to be exhaustive, we shall present in this paper an historical account of these theorems and their proofs, and clarify their mutual relation. We will explain how a suitable convention can help understand the original statement of Descartes and greatly simplify its proof, as well as the proofs of the above-mentioned theorems. With the exception of the proof of Fourier's theorem and its generalizations, which run on rudiments of infinitesimal calculus (Taylor's theorem), the proposed demonstrations are so short and elementary they could be taught at the undergraduate level.