It is well known that two non intersecting circles in $R^2$ have four common tangents, and the proof for that is quite easy (either by direct computation or by intersection theory, but then one has to consider $RP^2$ or $CP^2$). It is also known that for two generic parabolas in $R^2$ there are max. three common tangents. Can one prove it via intersection theory (or via other methods of algebraic geometry), without calculating directly?
Thank you, Dan