# analyticSpread -- Compute the analytic spread of a module or ideal

## Synopsis

• Usage:
• Inputs:
• M, , or an ideal
• f, , an optional element, which is a non-zerodivisor such that M[f-1] is a free module when M is a module, an element in M when M is an ideal
• Optional inputs:
• Outputs:
• an integer, the analytic spread of a module or an ideal M

## Description

The analytic spread of a module is the dimension of its special fiber ring. When I is an ideal (and more generally, with the right definitions) the analytic spread of I is the smallest number of generators of an ideal J such that I is integral over J. See for example the book Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, 2006, by Craig Huneke and Irena Swanson.

 ```i1 : R=QQ[a..h] o1 = R o1 : PolynomialRing``` ```i2 : M=matrix{{a,b,c,d},{e,f,g,h}} o2 = | a b c d | | e f g h | 2 4 o2 : Matrix R <--- R``` ```i3 : analyticSpread minors(2,M) o3 = 5``` ```i4 : specialFiberIdeal minors(2,M) o4 = ideal(Z Z - Z Z + Z Z ) 2 3 1 4 0 5 o4 : Ideal of QQ[Z , Z , Z , Z , Z , Z ] 0 1 2 3 4 5``` ```i5 : R=QQ[a,b,c,d] o5 = R o5 : PolynomialRing``` ```i6 : M=matrix{{a,b,c,d},{b,c,d,a}} o6 = | a b c d | | b c d a | 2 4 o6 : Matrix R <--- R``` ```i7 : analyticSpread minors(2,M) o7 = 4``` ```i8 : specialFiberIdeal minors(2,M) 2 2 o8 = ideal (Z Z - Z Z + Z Z , Z - Z Z - Z Z - Z + Z Z + Z Z ) 2 3 1 4 0 5 1 0 2 0 3 4 2 5 3 5 o8 : Ideal of QQ[Z , Z , Z , Z , Z , Z ] 0 1 2 3 4 5```