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Given any matrix of univariate polynomials of degree $\leq n^{O(1)}$ then prove that the coefficent of $x^i$ in the determinant of the matrix is in $GapL$

Hint: Use Mahajan-Vinay's result of converting determinant into a arithmetic branching program.

Now i thought that first we replace every polynomial in the matrix entry with new variables. So the $(i,j)$th entry of the matrix is replaced by a new variable $x_{i,j}$. Now using mahajan vinay we can compute the determinant of this new matrix using a arithmetic branching program. Now in that $ABP$ i am replacing all the $x_{i,j}$ with the original entry of the the given matrix. Now in the $ABP$ all the edge weights are univariate polynomial. Now for any edge if the edge weight polynimial has degree $k$ then we branch that edge into $k+1$ edges where the $i$th branch has weight $x^i$ multiplied by coefficient of $x^i$ in that polynomial. We will do this process for each edge. Now each edge has weight some constant multiplied by some power of $x$. Now let any edge has weight $cx^k$. Then we will break that edge into consecutive $k$ edges where the first edge has weight $kx$ and the next ones has weight $x$.

But i am not getting anywhere. I wanted finally to have like sum of $i$ length paths from source to target in the $ABP$ is the coefficient of $x^i$ in the determinant and if finding number of $i$ length paths in a graph is in $GapL$ then we are done.

Any ideas what should i do.

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Theorem 3.2 in this paper: https://doi.org/10.1007/s00224-003-1077-7 says this:

Let B be an n × n matrix, whose entries are each polynomials of degree n in Z[x], where the coefficients of each polynomial Bi, j are computable in GapL. Then the coefficients of the polynomial det(B) are computable in GapL.

The proof presented there seems convincing to me. (But then -- I am one of the authors of the paper.)

This would seem to answer your question.

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