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Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the polynomial can be compactly represented compact product forms.

Given $2n$ homogeneous polynomials in $2n$ variables $$f_j(x_1,\dots,x_n,y_j)=\prod_{i\in\mathcal T_j}(x_i-y_j)$$ $$g_i(x_i,y_1,\dots,y_n)=\prod_{j\in\mathcal U_i}(x_i-y_j)$$ where $\mathcal T_j,\mathcal U_i$ are subsets of $\{1,\dots,n\}$ expressing the resultant for all the $f_j$'s and the resultant for all the $f_j,g_i$'s as determinant of Macaulay matrices directly takes exponential time for both. Since the polynomials are structured is there a closed form for the two resultants or at least can these be computable by a determinant of polynomial size?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the polynomial can be compactly represented compact product forms.

Given $2n$ homogeneous polynomials in $2n$ variables $$f_j(x_1,\dots,x_n,y_j)=\prod_{i\in\mathcal T_j}(x_i-y_j)$$ $$g_i(x_i,y_1,\dots,y_n)=\prod_{j\in\mathcal U_i}(x_i-y_j)$$ where $\mathcal T_j,\mathcal U_i$ are subsets of $\{1,\dots,n\}$ expressing the resultant for all the $f_j$'s and the resultant for all the $f_j,g_i$'s as determinant of Macaulay matrices directly takes exponential time for both. Since the polynomials are structured is there a closed form for the two resultants or at least can the latter resultant of both $f_j,g_i$ be computable by a determinant of polynomial size?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the polynomial can be compactly represented compact product forms.

Given $2n$ homogeneous polynomials in $2n$ variables $$f_j(x_1,\dots,x_n,y_j)=\prod_{i\in\mathcal T_j}(x_i-y_j)$$ $$g_i(x_i,y_1,\dots,y_n)=\prod_{j\in\mathcal U_i}(x_i-y_j)$$ where $\mathcal T_j,\mathcal U_i$ are subsets of $\{1,\dots,n\}$ expressing the resultant for all the $f_j$'s and the resultant for all the $f_j,g_i$'s as determinant of Macaulay matrices directly takes exponential time for both. Since the polynomials are structured is there a closed form for the two resultants?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the polynomial can be compactly represented compact product forms.

Given $2n$ homogeneous polynomials in $2n$ variables $$f_j(x_1,\dots,x_n,y_j)=\prod_{i\in\mathcal T_j}(x_i-y_j)$$ $$g_i(x_i,y_1,\dots,y_n)=\prod_{j\in\mathcal U_i}(x_i-y_j)$$ where $\mathcal T_j,\mathcal U_i$ are subsets of $\{1,\dots,n\}$ expressing the resultant for all the $f_j$'s and the resultant for all the $f_j,g_i$'s as determinant of Macaulay matrices directly takes exponential time for both. Since the polynomials are structured is there a closed form for the two resultants or at least can these be computable by a determinant of polynomial size?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $$ polynomial. In certain cases the polynomial can be compactly represented compact product forms.

Given $2n$ homogeneous polynomials in $2n$ variables $$f_j(x_1,\dots,x_n,y_j)=\prod_{i\in\mathcal T_j}(x_i-y_j)$$ $$g_i(x_i,y_1,\dots,y_n)=\prod_{j\in\mathcal U_i}(x_i-y_j)$$ where $\mathcal T_j,\mathcal U_i$ are subsets of $\{1,\dots,n\}$ expressing the resultant for all the $f_j$'s and the resultant for all the $f_j,g_i$'s as determinant of Macaulay matrices directly takes exponential time for both. Since the polynomials are structured is there a closed form for the two resultants?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the polynomial can be compactly represented compact product forms.

Given $2n$ homogeneous polynomials in $2n$ variables $$f_j(x_1,\dots,x_n,y_j)=\prod_{i\in\mathcal T_j}(x_i-y_j)$$ $$g_i(x_i,y_1,\dots,y_n)=\prod_{j\in\mathcal U_i}(x_i-y_j)$$ where $\mathcal T_j,\mathcal U_i$ are subsets of $\{1,\dots,n\}$ expressing the resultant for all the $f_j$'s and the resultant for all the $f_j,g_i$'s as determinant of Macaulay matrices directly takes exponential time for both. Since the polynomials are structured is there a closed form for the two resultants?