Properties

BaseField(A) : FldAC -> Fld

Return the base field over which A is defined.

Rank(A) : FldAC -> RngIntElt

Return the current rank of A, that is, the number of variables which currently define A. This can increase by the construction of new roots, or decrease by pruning (see Roots and Prune respectively below), so the return value of this function is variable.

Degree(A, v) : FldAC, RngIntElt -> RngIntElt

Given an algebraically closed field A of rank r and an integer v in the range 1≤v≤r, return the current degree of the defining polynomial for variable v. The return value of this function is variable, as A may be simplified between invocations, making the defining polynomial for v have smaller degree.

Degree(A) : FldAC -> RngIntElt

Return the current absolute degree of A, that is, the degree over its base field. This necessitates the simplification of A (see Simplify below), so may be very time consuming. The return value varies only when new roots of polynomials over the field are computed, but until then, the return value is invariable (as the field will remain simplified, even if Prune or Absolutize is called -- see below).

AffineAlgebra(A) : FldAC -> RngMPolRes

QuotientRing(A) : FldAC -> RngMPolRes

Return the affine algebra (or multivariate quotient ring) R which currently represents A. The quotient relations of R consist of the defining polynomials A, and one may coerce between A and R, but note that the variable numbers are inverted. The reason for this is that for the system to work, the first root A.1 must be the smallest variable with respect to the lexicographical order in the corresponding affine algebra R, so that reductions modulo the Gröbner basis of relations are in the correct form.

Note also that if A changes in any way (whether from simplification or by pruning), then the affine algebra R of course stays the same and will not be comparable with the new form of A if A has a different number of variables than before.

Ideal(A) : FldAC -> RngMPol

Return the ideal of defining polynomials currently defining A. This is simply equivalent to DivisorIdeal(AffineAlgebra(A)). See the relevant comments for the function AffineAlgebra.