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In this paper, I shall discuss Leibniz’s considerations on the problem of exactness, as they can be reconstructed from published and unpublished letters, notes, drafts and sketches he did while Paris between 1673 and 1676. Leibniz’s critical target was Descartes’ Geometrie (1637). Descartes had managed to include into geometry all algebraic curves on the ground that they can be constructed by one continuous motions. Few rogue elements, called “mechanical curves” did not comply with this criterion of exactness, and therefore were not allowed in solving geometrical problems: these were, for example, the Archimedean spiral, the quadratrix and the cycloid. From the beginning of his mathematical studies, in 1673, and throughout his career, Leibniz set the task to reform the Cartesian criterion of exactness. Leibniz objected that Descartes’ criterion is not tenable, because there exist curves constructible by one continuous motion which cannot be associated to any algebraic equation. In this paper, I shall discuss several examples of such geometrical, non-algebraic curves: evolutes and involutes, the parabolic trochoid (the curve generated by the rolling of a parabola along one of its tangents) and the cycloid. The key to reform the Cartesian demarcation was, for Leibniz, to admit within geometry curve constructions obtained by strings that could be bent into arcs. In this way, the previously mentioned curves resulted geometrical insofar as they could be generated by one and continuous motion.

In this paper, we present a symbolic algorithm to compute the topology of a plane curve. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. The novelty of this paper is that we use a technique of interval polynomials to solve the system $\big\{f\alpha,\,y=\frac{\partial f}{\partial y}\alpha,\,y=0\big\}$ and at the same time, get the simple roots of fα, y=0 on the α fiber. It greatly improves the efficiency of the lifting step since we need not compute the simple roots of fα, y=0 any more. After the topology is computed, we use a revised Newton's method to compute the visualization of the plane algebraic curve. We ensure that the meshing is topologically correct. Many nontrivial examples show our implementation works well.

The impression is given in several places that few people responded quickly to Riemann ’s work, and those that did so with a view to finding fault with it, even arguing that it should be dismissed. It is true that some mathematicians began their papers by lamenting that nothing seems to have been done, but in fact his papers, with a few notable exceptions, drew a considerable amount of attention, as this chapter discusses.

2D regular expressions represent rational relations over two alphabets. In this paper we study the average state complexity of partial derivative standard transducers (\(\mathcal {T}_{\text {PD}}\)) that can be defined for (general) 2D expressions where basic terms are pairs of ordinary regular expressions (1D). While in the worst case the number of states of \(\mathcal {T}_{\text {PD}}\) can be \(O(n^2)\), where n is the size of the expression, asymptotically and on average that value is bounded from above by \(O(n^{\frac{3}{2}})\). Moreover, asymptotically and on average the alphabetic size of a 2D expression is half of the size of that expression. All results are obtained in the framework of analytic combinatorics considering generating functions of parametrised combinatorial classes defined implicitly by algebraic curves. In particular, we generalise the methods developed in previous work to a broad class of analytic functions.

This paper uses a formulation of the motion of the planar four-bar mechanism in the projective three-space of planar kinematic mapping and an algebraic version of the input-output relation is derived. This fourth order algebraic curve describes a new convenient form of the Freudenstein equation. Different geometric properties of the curve, independent of the design parameters, are carried out and their impact on the topology of the mechanism is shown. Furthermore the coefficients of this algebraic input-output curve are interpreted geometrically in the design parameter space which yields regions of parameter sets to the different topologies of four-bar mechanisms.

The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over \(\mathbb {F}_{q}\,\). Research on bounds for A(q) is closely connected with the so-called asymptotic main problem in Coding Theory. In this paper, we study some generalizations of this number for non-irreducible curves, their connection with A(q) and their application in Coding Theory. We also discuss the possibility of constructing codes from non-irreducible curves, both from theoretical and practical point of view.

Kempe’s universality theorem introduced in 1876 has recently been proven to ensure that given any algebraic curve a mechanism exists that traces the curve. In this paper, we present two methods to simplify Kempe’s linkages. One method uses gear trains, differentials and belt drives to replace his multiplicator, additor and translator linkages. A second method uses the Scotch yoke mechanism and a summing belt drive to generate a mechanical Fourier series that traces the curve. Examples are provided that demonstrate the two approaches.

In his 1876 paper [89], Harnack examined the following question: given a real algebraic curve of degree d in the projective plane, how many connected components can it have? He proved

We saw in Chap. 11 that Abel had a very ambitious program aiming to study all possible relations between abelian integrals. Even if he did not always completely prove them, he discovered many theorems concerning those relations. For instance, in [115] Kleiman lists and sketches modern proofs of the theorems he encountered in Abel’s paper [1]. One of the most famous such theorems, which is explained in nearly every textbook on algebraic curves and Riemann surfaces, is the following one

Since the beginnings of the development of the theory of algebraic surfaces, geometers have tried to prove a generalization of the Riemann–Roch theorem for algebraic curves (see Zariski’s account in [195, Chap. IV], as well as Gray ’s article [84]). Max Noether formulated such a generalization in his 1886 paper [144]. But he could only prove an inequality (I use the notations and the explanations from [32, Chaps. 7 and 35])

Riemann continued to develop his own ideas, extending them to multiply-connected domains defined by algebraic curves. He presented them in public when he lectured on complex functions, in particular elliptic and Abelian functions , in 1855/56 and again in 1861/62, and he published them in his remarkable paper on abelian functions in 1857.

Let \(\chi \) be a smooth, geometrically irreducible and projective curve over a finite field \({\mathbb F}_q\) of odd characteristic. The L-polynomial \(L_\chi (t)\) of \(\chi \) determines the number of rational points of \(\chi \) not only over \({\mathbb F}_q\) but also over \({\mathbb F}_{q^s}\) for any integer \(s \ge 1\). In this paper we determine L-polynomials of a class of such curves over \({\mathbb F}_q\).