Singular

D.12.6 rootsmr_lib

Library:

rootsmr.lib

Purpose:

Counting the number of real roots of polynomial systems

Author:

Enrique A. Tobis, [email protected]

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.12.6.1 nrRootsProbab		Number of real roots of 0-dim ideal (probabilistic)
D.12.6.2 nrRootsDeterm		Number of real roots of 0-dim ideal (deterministic)
D.12.6.3 symsignature		Signature of the symmetric matrix m
D.12.6.4 sturmquery		Sturm query of h on V(I)
D.12.6.5 matbil		Matrix of the bilinear form on R/I associated to h
D.12.6.6 matmult		Matrix of multiplication by f (m_f) on R/I in the basis B
D.12.6.7 tracemult		Trace of m_f (B is an ordered basis of R/I)
D.12.6.8 coords		Coordinates of f in the ordered basis B
D.12.6.9 randcharpoly		Pseudorandom charpoly of univ. projection, n optional
D.12.6.10 verify		Verifies the result of randcharpoly
D.12.6.11 randlinpoly		Pseudorandom linear polynomial, n optional
D.12.6.12 powersums		Powersums of the roots of a char polynomial
D.12.6.13 symmfunc		Symmetric functions from the powersums S
D.12.6.14 univarpoly		Polynomial with coefficients from l
D.12.6.15 qbase		Like kbase, but the monomials are ordered