How to find Hilbert basis of these equations?
$x_1 + x_2 + x_3 = x_4 + x_ 5 + x_6 = x_7 + x_8 + x_9 $
$x_1 + x_4 + x_7 = x_2 + x_ 5 + x_8 = x_3 + x_6 + x_9 $
2026-03-25 20:58:02.1774472282
Find Hilbert Basis of Magic Square Equations
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The first $3$ polynomials are already orthogonal $$p_1=x_1+x_2+x_3$$ $$p_2=x_4+x_5+x_6$$ $$p_3=x_7+x_8+x_9$$ The space only has $5$ dimensions. As was pointed out, linear algebra suffices to determine $2$ more orthogonal polynomials $$p_4=2x_1-x_2-x_3+2x_4-x_5-x_6+2x_7-x_8-x_9$$ $$p_5=x_2-x_3+x_5-x_6+x_8-x_9$$ Now we verify that the polynomials in question are in the space spanned by the $p_i$ $$x_1+x_4+x_7=\frac{1}{3}(p_1+p_2+p_3+p_4)$$ $$x_2+x_5+x_8=\frac{1}{8}(2p_1+2p_2+2p_3-p_4+4p_5)$$ $$x_3+x_6+x_9=\frac{1}{8}(2p_1+2p_2+2p_3-p_4-4p_5)$$