The Bezout number for linear piecewise algebraic curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper discusses the Bezout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, on regular triangulations. We give an upper bound of the Bezout number for linear piecewise algebraic curves (BN(1,0;1,0;Δ)BN(1,0;1,0;Δ)) on the triangulation with an odd interior vertex. For the triangulations which satisfy a vertex coloring condition, we compute the exact value of the Bezout number BN(1,0;1,0;Δ)BN(1,0;1,0;Δ).