Elements of a local polynomial ring are multivariate polynomials just like the usual (global) multivariate polynomials, except that the monomials are sorted (again with the greatest first) with respect to a local monomial order, which is in general the negation of a standard global monomial order. Thus the monomial 1 is less than all other monomials and the polynomials are like multivariate formal power series (written, for example, as 1 + x + x2y + y4). But most arithmetic-like operations allowed for global polynomials also carry over automatically for elements of a local polynomials so we will not list them in detail in this chapter (see Chapter MULTIVARIATE POLYNOMIAL RINGS).
Note that in the strict mathematical definition of R=K[x1, ..., xn]< x1, ..., xn >, elements of R may have non-trivial denominators, but this is currently not supported in Magma: the elements in Magma must always be strict polynomials. The main purpose of supporting such rings is for standard bases of ideals (see below), and this restriction does not matter there, since units are automatically removed from the elements of a standard basis.
We now describe the current local monomial orders available in Magma. First the reader should see Section Representation and Monomial Orders for the fundamental points about (global) monomial orders for multivariate polynomial rings.
The fundamental difference in the local case is that for a local polynomial ring R of rank n, the monomial order is the negation of a global monomial order. More precisely, let M be the monomials of R. A local monomial ordering on M is a total order < on M such that s ≤1 for all s ∈M, s ≤t implies su ≤tu for all s, t, u ∈M, and M is a well-ordering (every non-empty subset of M possesses a minimal element w.r.t. <). See [CLO98, Sec. 4.3], [DL06, Sec. 9.1], or [GP02, Sec. 1.2] for more information.
We now list each of the monomial orders available in Magma (these will be expanded in future versions). As in the global case, we suppose that s and t are monomials from a ring R of rank n. Any order on the monomials is then fully defined by just specifying exactly when s < t with respect to that order. In the following, the argument(s) are described for an order as a list of expressions; that means that the expressions (without the parentheses) should be appended to any base arguments when any particular intrinsic function is called which expects a monomial order.
Definition: s < t iff there exists 1 ≤i ≤n such that all of the j-th exponents of s and t are equal for i < j ≤n, but the i-th exponent of s is greater than the i-th exponent of t. The order is specified by the argument ("llex").
This order is the negation of the global lexicographical order, but with the reverse order for the variables. Thus the i-th variable is greater than the (i + 1)-th variable for 1 ≤i < n so the first variable is the greatest variable.
Definition: s < t iff the total degree of s is greater than the total degree of t or the total degree of s is equal to the total degree of t and s > t with respect to the (glocal) lexicographical order. The order is specified by the argument ("lglex").
This order is the negation of the global glex order.
Definition: s < t iff the total degree of s is greater than the total degree of t or the total degree of s is equal to the total degree of t and s < t with respect to the (global) lexicographical order applied to the exponents of s and t in reverse order. The order is specified by the argument ("grevlex").
This order is the negation of the global grevlex order.