Closed formula for univariate subresultants in multiple roots
Let $K$ be a field and $A,B\subset K$ be two sets of cardinalities $m$ and $n$, respectively. Then, in 1840, Sylvester introduced the Sylvester single sum as \[ \mathrm{Syl}_d(A,B)(x)=\sum_{A'\subset A, |A'|=d}\frac{R(A\setminus A',B)R(x,A')}{R(A',A\setminus A')} \] where $0 \le d \le m$ and $R(X,Y)=\prod_{x\in X, y\in Y}(x-y)$. If $f(x)=\prod_{a\in A}(x-a)$ and $g(x)=\prod_{b\in B}(x-b)$ then the Sylvester subresultant of $f$ and $g$ for $d$, up to multiplying by a constant in $\{1,-1\}$, is equal to $\mathrm{Syl}_d(A,B)(x)$. \par In this paper, the authors generalize the Sylvester single sum to multilists as follows: Keeping the above notations, let $\bar{A}\subset A$ and $\bar{A}\subset A$ be the sets of distinct elements in $A$ and $B$, respectively. Let $m'=m-|\bar{A}|$ and $n'=n-|\bar{B}|$. Then, $\mathrm{SylM}_d(A,B)(x)$ is defined to be \[ (-1)^{m'(m-d)}\sum_{A'\subset \bar{A}, |A'|=d-m'}\sum_{B'\subset\bar{B}, |B'|=m'}\frac{R(A\setminus \bar{A},\bar{B}\setminus B')R(\bar{A}\setminus A',B\setminus B')R(x,A')R(x,B')}{R(A',\bar{A}\setminus A')R(B',\bar{B}\setminus B')} \] where $m'+n'\le d \le \min\{m,n\}$. In the case that $m=n$ then we consider $d<m$. It is shown that the $d$-th Sylvester subresultant of $f$ and $g$, up to multiplying by a constant in $\{1,-1\}$, is equal to $\mathrm{Syl}_d(A,B)(x)$.
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