I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by one square-free univariate polynomial $g$ and two rational endpoints. All the coefficients are integers.
I more-or-less understand what to do to find out the sign of univariate polynomial $f$ evaluated over single algebraic number $a$: I compute generalized Sturm sequence or pseudo-remainder sequence of $f$ and $g$, evaluate the sequence twice over each of the endpoints of $a$ and count the number of sign changes.
But I don't understand how to compute the PRS of a bivariate polynomial and an univariate polynomial. The problem is that I don't know how to find the reminder of a bivariate and an univariate polynomial. Is there a way do divide bivariate polynomial by univariate polynomial? Should I perhaps somehow (how?) "transform" the bivariate polynomial into univariate one and do as before?
I also read that it should be possible to consider a bivariate polynomial ($f \in Z[x, y]$) as univariate polynomial in ($f \in (Z[y])[x]$) by means of "binary segmentation". Is this how should I proceed? How does it work? To be honest I really don't get how to "consider" a bivariate polynomial as a univariate polynomial. Or perhaps there is another way how to compute the sign of bivariate polynomial over two algebraic numbers?