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# localRingValuation -- The valuation defined by a local ring.

## Synopsis

• Usage:
v = localRingValuation(R)
• Inputs:
• R, , the ring whose maximal ideal determines the order
• Outputs:
• v, an instance of the type Valuation, local ring valuation using a local ring R

## Description

This function constructs a valuation which returns the largest power of the maximal ideal of R that contains the input to the valuation.

 i1 : R = QQ[x,y]; i2 : I = ideal(x,y); o2 : Ideal of R i3 : S = R_I o3 = S o3 : LocalRing, maximal ideal (x, y) i4 : localVal = localRingValuation(S) o4 = valuation from S to ZZ o4 : Valuation i5 : localVal(1 + x + y) o5 = 0 i6 : localVal(x^4 + x^2*y^2 + x^7 + y^3) o6 = 3 i7 : localVal(x^2 + x*y + y^2) o7 = 2

This valuation may be applied to elements of the fraction field of R, where the value of the valuation is the difference between the valuations of the numerator and denominator.

 i8 : localVal(1/(x^2+x*y+y^3)) o8 = -2