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README for "Rolle". Created by Christopher W. Brown, 6/28/2004 0. INSTALL: You must have a QEPCAD B (version 1.28 or beyond) installation to compile Rolle. Make sure that the environment variable "qe" is set to the base path of your qepcad installation and "saclib" to te base path of your saclib installation (they both ought to be if you've installed qepcad). Type "make", and you ought to have a program called "rolle" that's ready to use. 1. The "rolle" system is a very slight modification of QEPCAD B designed to efficiently construct all generic sign-stack sequences for a given polynomial. 2. The sign stack sequence problem comes from the following thesis: Bruce Anderson, May 1992 Advisor: Moss Sweedler, Cornell University "Signed Sequences and Rolle's Restrictions: Why Not All Real Differentiable Functions and Polynomials Satisfying Rolle's Theorem Are Constructible" Abstract: We investigate which sign sequences can be generated by a single univariate polynomial, and show that there are restrictions other than Rolle's theorem on polynomials. We then prove a fifth order Rolle's theorem, showing that an arrangement of roots satisfying the classical Rolle's theorem is not constructable by a real differentiable function. 3. Documentation for the program may be found at the top of MAIN.cc. 4. If f is a monic nth degree polynomial in x, the "sign stack sequence" for f is a sequence of (n+1)-tuples. Each tuple represents the signs of (f,f',f'',..,f^(n)) at a point. The sequence gives all the the distinct sign-tuples taken by f a x goes from -infinity to +infinity, excluding points at which f or any of its derivatives are zero. 5. If f has parameters in its coefficients, we may ask what sign-stack sequences are realized by f as the parameters vary over all their possible values. 6. Here we're interested only in situations in which f is "generic", which means that there are no pairwise common zeros amongst f and its derivatives.