### D.12.7 rootsmr_lib

Library:
  rootsmr.lib
Purpose:
  Counting the number of real roots of polynomial systems
Author:
  Enrique A. Tobis, etobis@dc.uba.ar

Overview:
  Routines for counting the number of real roots of a multivariate polynomial system. Two methods are implemented: deterministic computation of the number of roots, via the signature of a certain bilinear form (nrRootsDeterm); and a rational univariate projection, using a pseudorandom polynomial (nrRootsProbab). It also includes a command to verify the correctness of the pseudorandom answer.

References:
  Basu, Pollack, Roy, "Algorithms in Real Algebraic Geometry", Springer, 2003.

Procedures:

> D.12.7.1 nrRootsProbab |  | Number of real roots of 0-dim ideal (probabilistic)
> D.12.7.2 nrRootsDeterm |  | Number of real roots of 0-dim ideal (deterministic)
> D.12.7.3 symsignature |  | Signature of the symmetric matrix m
> D.12.7.4 sturmquery |  | Sturm query of h on V(I)
> D.12.7.5 matbil |  | Matrix of the bilinear form on R/I associated to h
> D.12.7.6 matmult |  | Matrix of multiplication by f (m_f) on R/I in the basis B
> D.12.7.7 tracemult |  | Trace of m_f (B is an ordered basis of R/I)
> D.12.7.8 coords |  | Coordinates of f in the ordered basis B
> D.12.7.9 randcharpoly |  | Pseudorandom charpoly of univ. projection, n optional
> D.12.7.10 verify |  | Verifies the result of randcharpoly
> D.12.7.11 randlinpoly |  | Pseudorandom linear polynomial, n optional
> D.12.7.12 powersums |  | Powersums of the roots of a char polynomial
> D.12.7.13 symmfunc |  | Symmetric functions from the powersums S
> D.12.7.14 univarpoly |  | Polynomial with coefficients from l
> D.12.7.15 qbase |  | Like kbase, but the monomials are ordered

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