Degree, multiplicity, and inversion formulas for rational surfaces using \(u\)-resultants (Q1195068)

Given an algebraic surface, the basic problems of identification of base points, minimal degree, generic representation, etc. are solved in algebraic geometry by intersecting the surface with a linear pencil. The authors note that if the surface is given parametrically in projective (3-)space by \(x_ i=f_ i(s_ k)\) then the intersection of the surface with a line given as intersection of two planes gives two equations \(\sum a^{(j)}_ if(s_ 1,s_ 2,s_ 3)=0\), (\(j=1,2\)). The adjunction of a third \(\sum u_ ks_ k=0\) with indeterminates \(u_ k\) allows one to compute the resultant which can be used to solve problems one is interested in \{unless one runs into a surface with infinitely close singularities, the problem that is unsolvable with methods of enumerative algebraic geometry\}. However, in the untractable cases the resultant will vanish and therefore it is a very safe indicator of trouble and that is really the most important aspect for computational geometry.