# gaussianRing(ZZ) -- ring of Gaussian correlations on n random variables

## Synopsis

• Function: gaussianRing
• Usage:
gaussianRing n
• Inputs:
• n, an integer, the number of random variables
• Optional inputs:
• Coefficients => ..., default value QQ, optional input to choose the base field in markovRing or gaussianRing
• kVariableName => ..., default value "k", symbol used for indeterminates in a ring of Gaussian joint probability distributions
• lVariableName => ..., default value "l", symbol used for indeterminates in a ring of Gaussian joint probability distributions
• pVariableName => ..., default value "p", symbol used for indeterminates in a ring of Gaussian joint probability distributions
• sVariableName => ..., default value "s", symbol used for indeterminates in a ring of Gaussian joint probability distributions
• Outputs:
• a ring, a ring with indeterminates $s_{(i,j)}$ for $1 \leq i \leq j \leq n$

## Description

This function creates a polynomial ring with indeterminates $s_{(i,j)}$ for $1 \leq i \leq j \leq n$. The $s_{(i,j)}$ indeterminates in the gaussianRing are the entries in the covariance matrix of the jointly normal random variables.

 i1 : R = gaussianRing 5; i2 : gens R o2 = {s , s , s , s , s , s , s , s , s , s , s , s , 1,1 1,2 1,3 1,4 1,5 2,2 2,3 2,4 2,5 3,3 3,4 3,5 ------------------------------------------------------------------------ s , s , s } 4,4 4,5 5,5 o2 : List i3 : compactMatrixForm =false; i4 : covarianceMatrix R o4 = | s s s s s | | 1,1 1,2 1,3 1,4 1,5 | | | | s s s s s | | 1,2 2,2 2,3 2,4 2,5 | | | | s s s s s | | 1,3 2,3 3,3 3,4 3,5 | | | | s s s s s | | 1,4 2,4 3,4 4,4 4,5 | | | | s s s s s | | 1,5 2,5 3,5 4,5 5,5 | 5 5 o4 : Matrix R <--- R