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

## Synopsis

• Usage:
val = valuation(R, v)
• Inputs:
• Outputs:

## 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