degreesRing(ZZ) -- the ring of degrees

This function produces a Laurent polynomial ring in n variablesT_0, ... , T_{n-1} whose monomials are to be used to represent degrees in another ring with multi-degrees of length n. If n=1, then the variable has no subscript.

i1 : degreesRing 3 o1 = ZZ [T , T , T , MonomialOrder => RevLex, Inverses => true] 0 1 2 o1 : PolynomialRing

i2 : T_0 o2 = T 0 o2 : IndexedVariable

Notice that the variables in this ring are local variables, but the command use will make the variables globally available.

i3 : use degreesRing 3 o3 = ZZ [T , T , T , MonomialOrder => RevLex, Inverses => true] 0 1 2 o3 : PolynomialRing

i4 : T_0 o4 = T 0 o4 : ZZ [T , T , T , MonomialOrder => RevLex, Inverses => true] 0 1 2

Elements of this ring are used as variables for Poincare polynomials generated by poincare and poincareN as well as Hilbertseries.

The degrees ring is a Laurent polynomial ring, as can be seen by the option in the definition of the ring that says Inverses => true. The monomial ordering used in the degrees ring is RevLex so the polynomials in it will be displayed with the smallest exponents first, because such polynomials are often used as Hilbert series.