= Sylvester matrix
{c}
For <polynomials> $p,q$ of degrees $m,n$, the <Sylvester matrix> has $n$ shifted coefficient rows of $p$ and $m$ shifted rows of $q$, with coefficients in descending-power order. It represents, after transposition, the map $(A,B)\mapsto Ap+Bq$ for $\deg A<n$, $\deg B<m$. Its <determinant> is the <resultant>. Its kernel is nonzero exactly when a common <polynomial factor> exists: divide $p,q$ by that factor to obtain a nonzero cancelling pair; conversely, coprimality and $Ap=-Bq$ imply $p\mid B$, forcing the degree-bounded $B$ and then $A$ to vanish.
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