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Consider two polynomials $f,g\in\mathbb{F}_2$ of degree $O(2^n)$, with the property that they are extremely sparse (say, only $O(n)$ of the coefficients are non-zero). Is there a way to calculate their resultant that does not involve an exponential computation in $n$? The Sylvester matrix would be very structured (more than it is already), but I don't know if there are methods to exploit this structure to compute its determinant.

In a more general context, I'm using the resultant to check if $f$ and $g$ share a root. Is there another method to check this, considering the sparsity property?

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See the oevre of I. Emiris:

Emiris, Ioannis Z.; Pan, Victor Y., Improved algorithms for computing determinants and resultants, J. Complexity 21, No. 1, 43-71 (2005). ZBL1101.68981.

Jeronimo, Gabriela; Sabia, Juan, Sparse resultants and straight-line programs, ZBL06825747.

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