SINGULAR - Base Rings

Singular may compute with polynomial objects over the following types of base rings:

• polynomial rings: K[x],   x=(x₁,...,x_n)

• localizations of a polynomial ring at a maximal ideal: K[x]_<x>

• tensor products of some of the above.

• factor rings by an ideal of one of the above: K[x]/J or K[x]_<x>/J.

• non-commutative G-algebras, that is, quotients of the free associative non-commutative algebra in x by the two-sided ideal generated by the rewriting relations x_j x_i - C_ij x_i x_j - D_ij, where the C_ij are nonzero constants and D_ij are linear combinations of monomials in K[x] subject to certain conditions (examples of G-algebras include quasi-commutative polynomial rings, universal enveloping algebras of finite dimensional Lie algebras, positive (negative) parts of quantized enveloping algebras, some iterated Ore extensions, some non-standard quantum deformations, Weyl algebras and quantizations of Weyl algebras, Smith algebras, ...). See the section on PLURAL in the Online Manual for details.

• non-commutative GR-algebras, that is, quotients of G-algebras by two-sided ideals (e.g. exterior algebras, Clifford algebras, finite dimensional associative algebras given by structure constants, ...).

The ground field K may be

• the field of rational numbers Q (char 0);

• a finite field Z/pZ, ( p <= 2147483629 );

• a Galois field, that is, a finite fields with q=pⁿ elements (pⁿ < 2¹⁵);

• a transcendental extension K(A₁,...,A_m) of K=Q or Z/pZ;

• a primitive algebraic extension K[t]/MinPoly of K=Q or Z/pZ;

In the context of solving systems of polynomial computations, Singular may also compute with floating point (real or complex) numbers (with precision of k< 32768 valid decimal digits).