# inhomogeneousSystemOfParameters -- generates an inhomogeneous system of parameters

## Synopsis

• Usage:
inhomogeneousSystemOfParameters(I,A)
• Inputs:
• Optional inputs:
• Attempts => an integer, default value 100, number of attempts made to generate an inhomogeneous system of parameters
• Bound => an integer, default value 1, bound on the value of the random coefficients
• Sparseness => , default value .5, between 0 and 1 giving the frequency of the coefficients being equal to zero
• Maximal => , default value true, whether to insist on searching for a maximal inhomogeneous system of parameters
• Outputs:

## Description

Given a ring and an ideal, inhomogeneousSystemOfParameters attempts to generate an inhomogeneous system of parameters contained in I. The algorithm is based on one found in Chapter 5.5 of W. Vasconcelos' book: Computational Methods in Commutative Algebra and Algebraic Geometry.
 i1 : A = ZZ/5051[x, y, z]; i2 : I = ideal (x, x*y, y*z); o2 : Ideal of A i3 : X = inhomogeneousSystemOfParameters(I,A) o3 = | yz+x xy | 1 2 o3 : Matrix A <--- A i4 : isRegularSequence(X,A) o4 = true
Here are examples with optional inputs:
 i5 : A = ZZ/5051[x, y, z]; i6 : I = ideal (x, x*y, y*z); o6 : Ideal of A i7 : inhomogeneousSystemOfParameters(I,A,Attempts=>1,Bound=>100,Sparseness=>.9) --warning: no maximal regular sequence found o7 = 0 1 o7 : Matrix A <--- 0
Here are examples with the optional input Maximal => false:
 i8 : x = symbol x; y = symbol y; i10 : n = 2; i11 : A = ZZ/101[x_(1,1)..x_(n,n),y_(1,1)..y_(n,n)]; i12 : X = transpose genericMatrix(A,n,n); 2 2 o12 : Matrix A <--- A i13 : Y = transpose genericMatrix(A,y_(1,1),n,n); 2 2 o13 : Matrix A <--- A i14 : b = ideal(X*Y - Y*X); o14 : Ideal of A i15 : B = A/b; i16 : inhomogeneousSystemOfParameters(B,Attempts=>1,Maximal=>false) o16 = | -x_(2,1)-y_(1,2)+y_(2,1) x_(1,2)-y_(2,2) ----------------------------------------------------------------------- -x_(2,1)+x_(2,2)-y_(1,1)+y_(1,2) -x_(1,1)+x_(2,1) ----------------------------------------------------------------------- x_(1,1)-x_(2,1)-y_(1,1) x_(1,1)+x_(1,2) | 1 6 o16 : Matrix B <--- B

This symbol is provided by the package Depth.

## Ways to use inhomogeneousSystemOfParameters :

• "inhomogeneousSystemOfParameters(Ideal)"
• "inhomogeneousSystemOfParameters(Ideal,Ring)"
• "inhomogeneousSystemOfParameters(Ring)"

## For the programmer

The object inhomogeneousSystemOfParameters is .