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A few riddles behind Rolle's theorem

B. Shapiro·M. Shapiro
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Abstract

We call a smooth function of one variable a degree n pseudopolynomial if its n-th derivative has no (real) zeros. An n pseudopolynomial is called hyperbolic if it has exactly n simple zeros. In this short note we describe the necessary and sufficient conditions on the arrangements of 6 points consisting of 3 zeros of pseudopolynomials of degree 3, two zeros of their 1st derivatives and 1 zero of their second derivatives. Besides the standard Rolle's inequalities the restrictions include additional quadratic inequalities of geometric origin. This text is an easy reading accessible for undergraduates. We formulated also two questions not solved yet.

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B. Shapiro, M. Shapiro. 2005-12-08. A few riddles behind Rolle's theorem. https://arxiv.org/abs/math/0302215

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