Magma allows the explicit computation of the variety of an ideal when that ideal is zero-dimensional, i.e., the set of solutions to the simultaneous equations represented by the ideal. When an ideal is not zero-dimensional, the corresponding variety over the algebraic closure of the coefficient field is infinite and it is not possible to compute with such varieties within Magma.

Variety(I) : RngMPol -> [ ModTupFldElt ]

Variety(I, L) : RngMPol, Rng -> [ ModTupFldElt ]

    Digits: RngIntElt                   Default: 38

Given a zero-dimensional ideal I of a polynomial ring P whose order is of lexicographic type, return the variety of I over its coefficient field K as a sequence of vectors. Each vector is of length n, where n is the rank of P, and corresponds to an assignment of the n variables of P (in order) such that all polynomials in I vanish with this assignment. The ideal must be zero-dimensional so that the variety is known to be finite. If a superfield L of K is also given, the variety is computed over L instead, so the entries of the vectors lie in L. If the field over which the variety is computed is the free complex field, Magma uses a special root finding algorithm to ensure the precision of the results; in this case, the parameter Digits may be given (see the Roots intrinsic function in the Real and Complex Fields chapter).

VarietySequence(I, L) : RngMPol, Rng -> [ [ RngElt ] ]

    Digits: RngIntElt                   Default: 38

Given a zero-dimensional ideal I of a polynomial ring P whose order is of lexicographic type, return the variety of I over its coefficient field K as a sequence of sequences of elements of K. Each inner sequence is of length n, where n is the rank of P, and corresponds to an assignment of the n variables of P (in order) such that all polynomials in I vanish with this assignment. The ideal must be zero-dimensional so that the variety is known to be finite. If a superfield L of K is also given, the variety is computed over L instead, so the entries of the vectors lie in L. If the field over which the variety is computed is the free complex field, Magma uses a special root finding algorithm to ensure the precision of the results; in this case, the parameter Digits may be given (see the Roots intrinsic function in the Real and Complex Fields chapter).

Given a zero-dimensional ideal I of a polynomial ring P over a field K, return the size of the variety of I over the algebraic closure K' of K. The size is determined by finding the (prime) radical decomposition of I and placing each component of the decomposition into normal position so the size of the variety of the component over K' can be read off. Note that this function will usually be much faster than actually computing the variety of I over a suitable extension field of K.

> K<w> := GF(27);
> P<x, y> := PolynomialRing(K, 2);
> I := ideal<P | x^8 + y + 2, y^6 + x*y^5 + x^2>;
> Groebner(I);
> I;
Ideal of Polynomial ring of rank 2 over GF(3^3)
Lexicographical Order
Variables: x, y
Dimension 0
Groebner basis:
[
    x + 2*y^47 + 2*y^45 + y^44 + 2*y^43 + y^41 + 2*y^39 + 2*y^38 + 2*y^37 + 
        2*y^36 + y^35 + 2*y^34 + 2*y^33 + y^32 + 2*y^31 + y^30 + y^28 + y^27 + 
        y^26 + y^25 + 2*y^23 + y^22 + y^21 + 2*y^19 + 2*y^18 + 2*y^16 + y^15 + 
        y^13 + y^12 + 2*y^10 + y^9 + y^8 + y^7 + 2*y^6 + y^4 + y^3 + y^2 + y +
        2
    y^48 + y^41 + 2*y^40 + y^37 + 2*y^36 + 2*y^33 + y^32 + 2*y^29 + y^28 + 
        2*y^25 + y^24 + y^2 + y + 1
]
> V := Variety(I);
> V;
[
    (w^14 w^12),
    (w^16 w^10),
    (w^22  w^4)
]
> // Check that the original polynomials vanish:
> [
>    <x^8 + y + 2, y^6 + x*y^5 + x^2> where x is v[1] where y is v[2]: v in V
> ];
[ <0, 0>, <0, 0>, <0, 0> ]
> // Note that the variety of I would be larger over an extension field of K:
> VarietySizeOverAlgebraicClosure(I);
48