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

Synopsis

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

Ways to use primeConesOfSubalgebra:

For the programmer

The object primeConesOfSubalgebra is a method function.