ROLLE - Polynomial Sign-Stack Sequences

What is ROLLE?

ROLLE is a very slightly modified version of QEPCAD B that allows one to efficiently generate all "sign-stack sequences" realized by a polynomial.

How do I get/install ROLLE?

ROLLE itself is a slight modification of QEPCAD B. So, if you can get QEPCAD B installed, ROLLE should be no problem. This means that under Solaris and Linux things should be OK.

1. Download and install QEPCAD B version 1.38 or higher. (Instructions & Download)
2. Download ROLLE: rolle.tar.gz
3. Unzip and untar rolle.tar.gz.
4. Change to the rolle directory and type "make".
5. Hopefully the executable ./rolle is sitting in yourrolle directory waiting to be used!

More info ...

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.