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3 Polynomials
 3.1 The Floats pseudo-field
 3.2 Roots of polynomials
 3.3 LLL lattice reduction

3 Polynomials

3.1 The Floats pseudo-field

Polynomials with floating-point coefficients may be manipulated in GAP; though they behave, in subtle ways, quite differently than polynomials over rings.

The "pseudo-field" of floating-point numbers is an object in GAP, called FLOAT_PSEUDOFIELD. (It is not really a field, e.g. because addition of floating-point numbers in not associative). It may be used to create indeterminates, for example as

gap> x := Indeterminate(FLOAT_PSEUDOFIELD,"x");
x
gap> 2*x^2+3;
2.0*x^2+3.0
gap> Value(last,10);
203.0

3.2 Roots of polynomials

The Jenkins-Traub algorithm has been implemented, in arbitrary precision for MPFR and MPC.

Furthermore, CXSC can provide complex enclosures for the roots of a complex polynomial.

3.3 LLL lattice reduction

A faster implementation of the LLL lattice reduction algorithm has also been implemented. It is accessible via the commands FPLLLReducedBasis(m) and FPLLLShortestVector(m).

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