Given a polygonal curve P , a pointset S, and an ε > 0, we study the problem of finding a polygonal curve Q whose vertices are from S and has a Fréchet distance less or equal to ε to curve P . In this problem, Q must visit every point in S and we are allowed to reuse points of pointset in building Q. First, we show that this problem in NP-Complete. Then, we present a polynomial time algorithm f...

These tables record results on curves with many points over finite fields. For relatively small genus (0 ≤ g ≤ 50) and q a small power of 2 or 3 we give in two tables the best presently known bounds for Nq(g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field Fq of cardinality q. In additional tables we list for a given pair (g, q)...

We characterize the defining equation of a plane algebraic curve with exactly one rational place Q at infinity, then give a basis of L(mQ) with pairwise distinct pole orders at Q. The defining equation can be regarded as a generalization of the Weierstrass form of a hyperelliptic curve. In this informal note I give an English proof of the results of the Cab curve found by Miura [4, 5, 6]. Throu...

Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in R 2 with two independent approximation functions; that is if a certain sum converges then the set of all points (x, y) on the curve which satisfy simultaneously the inequalities qx < ψ1(q) and qy < ψ2(q) infinitely often has induced measure 0. This co...

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. This paper continues the study, begun in [FT] and [FGT], of curves over finite fields with many rational points, based on Stöhr-Voloch’s approach [SV] to the Hasse-Weil bound by way of Weierstrass Point Theory and Frobenius orders. Some of the results were announced in [T]. A...

We consider curves defined over small finite fields with points of large prime order over an extension field. Such curves are often referred to as Koblitz curves and are of considerable cryptographic interest. An interesting question is whether such curves are easy to construct as the target point order grows asymptotically. We show that under certain number theoretic conjecture, if q is a prim...