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explicit(WittPolynomialRing) -- Expresses a WittPolynomialRing as a finitely generated algebra over the integers.

Description

If R = (ZZ/p)[x_1,..., x_d] is a polynomial ring over a prime field ZZ/p, and $n \geq 1$ is an integer, then $W_n(R)$ can be identified with a certain $\mathbb{Z} / p^n$ subalgebra of a polynomial ring in d variables over $\mathbb{Z} / p^n$. This method finds the relations between these generators in order to express $W_n(R)$ as a finitely generated ZZ-algebra. Note that the number of generators and relations grows very fast with p and d.

When the method is applied when R is a polynomial ring over a finite but not prime field, the package essentially treats R as a quotient of a polynomial ring over its prime subfield.

i1 : R = (ZZ/2)[x];
i2 : WR = witt(2, R);
i3 : explicit(WR)

      ZZ[T        , T        ]
          {0, {1}}   {1, {1}}
o3 = --------------------------
                      2
     (4, 2T        , T        )
           {1, {1}}   {1, {1}}

o3 : QuotientRing
i4 : R = GF(4)[x];
i5 : WR = witt(2, R);
i6 : explicit(WR);

Ways to use this method:


The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/WittVectors/Documentation.m2:779:0.