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ComprehensiveSquareFreeFactorization.md

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ComprehensiveSquareFreeFactorization.md

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ComprehensiveSquareFreeFactorization

The ComprehensiveSquareFreeFactorization function will compute a complete case discussion of the square-free factorization of a parametric univariate polynomial.

Calling Sequence

ComprehensiveSquareFreeFactorization(p, v, R, options)
ComprehensiveSquareFreeFactorization(p, v, rs, R, options)
ComprehensiveSquareFreeFactorization(p, v, cs, R, options)
ComprehensiveSquareFreeFactorization(p, v, F, R, options)
ComprehensiveSquareFreeFactorization(p, v, F, H, R, options)

Input

Variable Description
p A parametric, univariate polynomial in v
v The variable that p1 and p2 are polynomials in
rs Regular system
cs Constructible set
F List of polynomials over R representing equality constraints on the parameters
H List of polynomials over R representing inequation constraints on the parameters
R Polynomial ring

Options

Option name Default value Description
'outputType' 'CS' When set to 'CS' or 'ConstructibleSet' the output will contain constructible sets, when set to 'RS' or 'RegularSystem', the output will contain regular systems.

Output

A list with elements of the form:

Output structure Options
[m_i, lp_i, rs_i] 'outputType' is 'RegularSystem' or 'RS'
[m_i, lp_i, cs_i] 'outputType' is 'ConstructibleSet' or 'CS'

Where m_i is a rational functions in the parameters (i.e. does not contain v as an indeterminate), lp_i is a list with elements of the form [q_j, n_j] such that p = m_i*product(q_j^n_j) and q_j are the square-free factors of the input polynomial for parameter values in the zero set of cs_i or rs_i.

Assumptions

  1. degree(p, mvar(R)) > 0

Example

with(RegularChains):
with(ParametricMatrixTools):

# Set up the polynomials with variable order x > a
R := PolynomialRing([x, a]):
p := (x+1)^2 * (x+a):

# Compute the square-free factorization
result := ComprehensiveSquareFreeFactorization(p, x, R):
nops(result);
    2
Display(result[1], R);
    [1, [[x+a, 1], [x+1, 2]], a-1 <> 0]
Display(result[2], R);
    [1, [[x+1, 3]], a-1 = 0]