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2 changed "invert a matrix" to "solve a matrix equation"

Your system of equations is of the form $Az = b$, where $z_{i(\alpha)} = x^\alpha$ and $i(\alpha)$ is the grlex index of

$\alpha \in$ $X_{n,k} \equiv$ {$\beta \in \mathbb{Z}^n: \sum_j \beta_j = k$}.

(See http://mathoverflow.net/questions/9477/uniquely-generate-all-permutations-of-three-digits-that-sum-to-a-particular-value/ for information on this index.)

Solving $Az = b$ is tantamount to solving the equations (if a solution exists), because it is trivial to obtain $x$ from $z$. So really you just need to invert solve a matrix equation once you sort through the indexing issues.

1

Your system of equations is of the form $Az = b$, where $z_{i(\alpha)} = x^\alpha$ and $i(\alpha)$ is the grlex index of

$\alpha \in$ $X_{n,k} \equiv$ {$\beta \in \mathbb{Z}^n: \sum_j \beta_j = k$}.

(See http://mathoverflow.net/questions/9477/uniquely-generate-all-permutations-of-three-digits-that-sum-to-a-particular-value/ for information on this index.)

Solving $Az = b$ is tantamount to solving the equations (if a solution exists), because it is trivial to obtain $x$ from $z$. So really you just need to invert a matrix once you sort through the indexing issues.