# isLogarithmic -- check if the given vector fields are logarithmic

## Synopsis

• Usage:
b=isLogarithmic(M,I)
b=isLogarithmic(V,I)
b=isLogarithmic(m,I)
• Inputs:
• M, , of vector fields
• V, , a column from a matrix of vector fields
• m, , of vector fields
• I, an ideal, the ideal
• Outputs:
• b, , whether the vector fields are in derlog(I)

## Description

Check if the vector field(s) given are in the module of logarithmic vector fields of I (see derlog). This function does not compute derlog(I), but instead applies the vector fields to the generators of I and checks if the result lies in I.

 i1 : R=QQ[x,y,z]; i2 : f=x*y-z^2; i3 : I=ideal (f); o3 : Ideal of R i4 : M=matrix {{x,2*z,2*z},{y,0,0},{z,y,x}}; 3 3 o4 : Matrix R <--- R i5 : applyVectorField(M,{f}) 2 o5 = {2x*y - 2z , 0, - 2x*z + 2y*z} o5 : List i6 : isLogarithmic(M,I) o6 = false i7 : isLogarithmic(M_{0,1},I) o7 = true i8 : isLogarithmic(derlog(I),I) o8 = true

• VectorFields -- a package for manipulating polynomial vector fields
• derlog -- compute the logarithmic (tangent) vector fields to an ideal
• applyVectorField -- apply a vector field to a function or functions

## Ways to use isLogarithmic :

• "isLogarithmic(Matrix,Ideal)"
• "isLogarithmic(Module,Ideal)"
• "isLogarithmic(Vector,Ideal)"

## For the programmer

The object isLogarithmic is .