# newField -- constructor of a vector field

## Synopsis

• Usage:
newField(n,d,varName)
• Inputs:
• n, an integer, number of variables minus one
• d, an integer, degree of the homogeneous polynomial coefficients
• varName, name of the generic scalar coefficients
• Outputs:
• an instance of the type DiffAlgField, a homogeneous vector field in (n+1)-dimensional affine space with generic scalar coefficients

## Description

This function defines homogeneous vector fields with generic scalar coefficients. By default, the affine coordinates will be x_0,...,x_n and the partial derivatives are denoted as ax_0,...,ax_n, respectively.

In this example we define a homogeneous vector field with linear polynomial coefficients in 3 variables. The scalar coefficients are chosen to be defined with the variable a. The index of the scalar coefficients will always start with 0.

 ```i1 : X = newField(2,2,"a") 2 2 2 2 o1 = (a x + a x x + a x + a x x + a x x + a x )ax + (a x + a x x + 0 0 3 0 1 9 1 6 0 2 12 1 2 15 2 0 1 0 4 0 1 ------------------------------------------------------------------------ 2 2 2 2 a x + a x x + a x x + a x )ax + (a x + a x x + a x + a x x + 10 1 7 0 2 13 1 2 16 2 1 2 0 5 0 1 11 1 8 0 2 ------------------------------------------------------------------------ 2 a x x + a x )ax 14 1 2 17 2 2 o1 : DiffAlgField``` ```i2 : ring X QQ[i] o2 = ------[][a , a , a , a , a , a , a , a , a , a , a , a , a , a , a , a , a , a ][x , x , x ][ax , ax , ax ] 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 2 0 1 2 i + 1 o2 : PolynomialRing```

## Caveat

The coefficient i is the imaginary unit.