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coneToValuation -- Convert a prime cone of a tropical ideal to a (quasi-)valuation

Description

This function constructs a valuation from the prime cone of a tropical variety. From a Subring and the rays of a prime cone of the kernel of its presentationRing, a (quasi-)valuation is constructed. A quasivaluation satisfies $\nu(fg)\geq\min\{\nu(f),\nu(g)\},$ where the inequality replaces the equality.

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 : C = 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 : val = coneToValuation(C#0, A)

o4 = valuation from QQ[p_0..p_3] to QQ^2

o4 : Valuation
 
i5 : use A#"presentationRing";
 
i6 : val(p_0^2 + p_1*p_2 - p_3^3)

o6 = |  -9 |
     | -18 |

o6 : Ordered QQ^2 module

See also

Ways to use coneToValuation :

For the programmer

The object coneToValuation is a method function.