Say we have $r$ linear polynomials $p_1,\ldots,p_r\in k[x_1,\ldots,x_n]$ which are linearly independent. How can we establish an automorphism of $k[x_1,\ldots,x_n]$ such that each $p_i$ goes to $x_i$? I know it has something to do with gaussian elimination, but I have no clue on how to proceed.
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What if $p_i = a_ix_1$? – Paul Frost Oct 18 '21 at 12:25
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I meant to say linear independent polynomials, sorry. – Darsen Oct 18 '21 at 14:40
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1Hint: Make the coefficients of $p_i$ into a matrix in $GL_n(k)$ and then construct an inverse homomorphism using the coefficients of the inverse of that matrix. – Daniel Oct 18 '21 at 19:54
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What about the zero degree terms? – Darsen Oct 18 '21 at 21:51