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valM -- Construct a valuation from a (quasi-)valuation

Synopsis

Description

Constructs a valuation from a (quasi-)valuation following the approach in Kaveh and Manon, 2019. In particular, the maximum quasi-valuation of all preimages of the input is taken as the valuation.

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;
 
i4 : v = coneToValuation(C#0, A);
 
i5 : vA = valM(R, v)

o5 = valuation from R to QQ^2

o5 : Valuation
 
i6 : use R;
 
i7 : vA(x_1^2 + x_2^2 + x_3^2)

o7 = | -2 |
     | -4 |

o7 : Ordered QQ^2 module
 
i8 : vA((x_1^2 - x_2^2)*(x_1^2 - x_3^2)*(x_2^2 - x_3^2))

o8 = |  -5 |
     | -10 |

o8 : Ordered QQ^2 module
 
i9 : vA(0_R)

o9 = infinity

o9 : InfiniteNumber

For elements not in A, the valuation returns unreliable results because the valuation does not come from a weight valuation on R

i10 : vA(x_2)

o10 = infinity

o10 : InfiniteNumber
 
i11 : vA(x_2^2)

o11 = | -2 |
      | -4 |

o11 : Ordered QQ^2 module
 
i12 : vA(x_2^3)

o12 = | -2 |
      | -4 |

o12 : Ordered QQ^2 module

References

K. Khovanskii and C. Manon. Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry.SIAM Journal on Applied Algebra and Geometry, 3(2), 2019.

See also

For the programmer

The object valM is a method function.