We address the description of tropicalization families rational varieties under parametrizations with prescribed support, via curve valuations. recover and extend results by Sturmfels, Tevelev Yu for generic coefficients, considering non-trivial denominator. The advantage our point view is that it can be generalized to deal non-generic parametrizations. provide a detailed analysis degree closed...

LetX denote the rational curve with n+1 nodes obtained from the Riemann sphere by identifying 0 with∞ and ζj with −ζj for j = 0, 1, . . . , n−1, where ζ is a primitive (2n)th root of unity. We show that if n is even, then X has no smooth Weierstrass points, while if n is odd, then X has 2n smooth Weierstrass points. C. Widland [14] showed that the rational curve with three nodes obtained from P...

New algorithm for evaluating a point on a Bézier curve and on a rational Bézier curve is given. The method has a geometric interpretation and a convex hull property. The computational complexity of the new algorithm is linear with respect to the degree of the considered curve, and its memory complexity is O(1).

Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theo...