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People who are actually supposed to answer this question are those who know about the Descartes' Rule of Signs, so I will not go about explaining it. The well-known proof for the Rule includes somewhat 6 lemmas and covers 7 pages or so, presented http://homepage.smc.edu/kennedy_john/POLYTHEOREMS.PDF [Broken], starting from (17). However, I came across a simpler proof that is presented http://www.math.tamu.edu/~rojas/wangdescartes.pdf [Broken], which covers somewhat 2 pages. My question is that is this last proof valid? I did not spot any mistakes so far, I am curious if you will.

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