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primeConesOfSubalgebra -- Finds the prime cones of the tropicalization of a given subalgebra or ideal.

Synopsis

• Usage:
C = primeConesOfSubalgebra(S)
C = primeConesOfIdeal(I)
• Inputs:
• Outputs:
• C, a list, containing the ray-generators of the prime cones.

Description

Let $I \subset k[x]$ be a prime ideal and let $C \subset \mathcal{T}(I)$ be an open cone in the tropicalization of $I$. This function returns all such $C$ where the initial ideal $\operatorname{in_{C}}(I)$ is a prime ideal. When the input is a Subring which is a domain, then $I$ is the kernel of the presentation map of $S$.

 i1 : R = QQ[x_1, x_2, x_3]; i2 : A = subring { x_1 + x_2 + x_3, x_1*x_2 + x_1*x_3 + x_2*x_3, x_1*x_2*x_3, (x_1 - x_2)*(x_1 - x_3)*(x_2 - x_3) }; i3 : primeConesOfSubalgebra A o3 = {| -3 22 |, | 22 -3 |, | -11 -2 |} | -6 -2 | | -2 -6 | | 1 19 | | 14 -3 | | -3 -9 | | 13 -6 | | -9 -3 | | -3 14 | | -10 -6 | o3 : List i4 : I = ideal(x_1*x_2+x_2^2+x_3^2); o4 : Ideal of R i5 : primeConesOfIdeal I o5 = {| -1 |} | 1 | | 0 | o5 : List

See also

• coneToValuation -- Convert a prime cone of a tropical ideal to a (quasi-)valuation
• valM -- Construct a valuation from a (quasi-)valuation

Ways to use primeConesOfSubalgebra :

• primeConesOfSubalgebra(Subring)

For the programmer

The object primeConesOfSubalgebra is .