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## Main Question or Discussion Point

**Problem:**Let R=Q[y] and suppose f,g \in F[x] both have degree 10 with respect to x and degree 6 with respect to y. Suppose h = gcd(f,g) has degree 4 with respect to x and degree 2 with respect to y. Derive an upper bound (as good as possible) on the number of distinct integers i such that gcd(f(x,i), g(x,i)) \in Q[x] has degree not equal to 4.

**Start of solution:**We can write h as h=p

_{4}(y)x

^{4}+p

_{3}(y)x

^{3}+p

_{2}(y)x

^{2}+p

_{1}(y)x + p

_{0}(y), where each p

_{i}is a polynomial of degree at most 2 in y. Then there exist at most 2 integers i that cause p

_{4}(y) to evaluate to zero, thus dropping the degree of h.

Also, evaluating f(x,i) is equivalent to computing f(x,y) mod (y-i). I know that there are results that say precisely when gcd(f mod i, g mod i) = gcd(f,g) mod i, and I suspect that these are required to find the remainder of the cases. But that's about as far as I can get. What results should I use to proceed?