A discrete monomial valuation v on R=K[X
1,...,X
n] is determined by the values v(X
j) of the indeterminates. This function takes as input the matrix V=(v
i(X
j)), whose rows correspond to the values of the indeterminates for for r valuations v
1, ...,v
r, with an additional column holding lower bounds w
1,...,w
r ∈ℤ. It returns the subalgebra S={f∈R: v
i(f)≥0, i=1,...,n}, the intersection of the valuation rings of the r valuations, and a system of generators of the S-submodule M={f∈R: v
i(f)≥w
i, i=1,...,n} over R, which consists of the elements whose i-th valuation is greater or equal to the i-th bound w
i. If w
i>=0 for all i, then M is an ideal in S.
This method can be used with the options
allComputations and
grading. The additional data can be accessed via the subalgebra in the
HashTable.
i1 : R=QQ[x,y,z,w];
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i2 : V=matrix({{0,1,2,3,4},{-1,1,2,1,3}});
2 5
o2 : Matrix ZZ <--- ZZ
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i3 : intersectionValRingIdeals(V,R)
No grading specified and cannot find one. Disabling some computations!
Warning: Cone could not compute everything, that it was asked for!
Missing: HilbertSeries
2 2 2 2 2 4 4 2 3
o3 = HashTable{module generators => {x*z , z , z*w, y z, x*y z, y w, x*y , y , y*w , w } }
subalgebra => MonomialSubalgebra{cache => CacheTable{...1...} }
2
generators => {x*y, y, x*w, x z, w, z, x*z}
ring => R
o3 : HashTable
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