# Coefficients of a determinant of a matrix of univariate polynomials is in $GapL$

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.