Ok, here goes.
Let's consider f(x,y) and g(x,y) as polynomials in x with coefficients in Q[y]. Then Computing gcd( f(x,i), g(y,i) ) is equivalent to computing gcd( f(x,y) mod (y-i), g(x,y) mod (y-i) ), where each y-i is a prime Q[y].
Thus, in order to determine when gcd( f(x,i), g(x,i) ) =...
And I stand by my objection to your answer, since (I believe) it is simply not the case that gcd(f(x,y),g(x,y)) evaluated at i is equal to gcd(f(x,i),g(x,i)) in general. (That is, the operations of taking a gcd and evaluating at a point only commute when certain conditions are satisfied. If I...
That does indeed seem to be the intuitive answer, however I suspect it is incorrect.
The reason is this: in general gcd(f(x) mod p, g(x) mod p) \neq gcd(f(x),g(x)) mod p. In particular, if p divides the gcd of the leading coefficients of f and g, then the two gcd's are different (and usually...
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...
Since 0 <= c_n * I(x-x_n) <= c_n, for all x \in [a,b], s_n(x) = \sum^n_{k=1}{c_k * I(x-x_k)} <= \sum^n_{k=1}{c_k} <= \sum^\infty_{k=1}c_k.
So for x \in [a,b], {s_n(x)} is monotone increasing and bounded above, thus convergent.
Since I(x-x_n) <= I(y-x_n), for all x<y, f is monotone...
Let I(x) (the unit jump function) be defined piecewise by I(x)=0 iff x<0 and I(x)=1 iff x>=0.
Let {x_n} be a countable subset of R and {c_n} a sequence of positive real numbers satisfying \sum{c_n}=1. Let f:R \to R be defined by f(x)=\sum{c_n * I(x - x_n)}.
Prove that...