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D.12.7 rootsmr_lib

Counting the number of real roots of polynomial systems
Enrique A. Tobis, etobis@dc.uba.ar

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.

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


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