 Degree bound in polynomial Bézout's identity

For two polynomials over the complex numbers $p,q$ of equal positive degree of a polynomial $n$ with no common root of a polynomial, there are unique polynomials $u,v$ of degree of a polynomial less than $n$ such that $up+vq=1$. To prove the bound, let $V_m$ be the vector space of polynomials of degree of a polynomial at most $m$, including the zero polynomial. The linear map $(a,b)\mapsto ap+bq$ from $V_{n-1}\times V_{n-1}$ to $V_{2n-1}$ has zero kernel of a linear map: if $ap=-bq$, each root of a polynomial of $p$, counted with its multiplicity of a root, must occur in $b$ because it does not occur in $q$, forcing $b=0$ by the degree of a polynomial bound, and then $a=0$. Both vector spaces have dimension $2n$, so the rank-nullity theorem makes the linear map a bijection, and $1$ has a unique preimage.