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

Synopsis

Usage:

val = valuation(R, v)
Inputs:
- R, a ring,
- v, an instance of the type Valuation,
Outputs:
- val, an instance of the type Valuation,

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.

Ways to use valM:

valM(Ring,Valuation)

For the programmer

The object valM is a method function.