Georg Grasegger

A solution method for autonomous first-order algebraic partial differential equations

In this paper we present a procedure for solving first-order autonomous algebraic partial differential equations in an arbitrary number of variables. The method uses rational parametrizations of algebraic (hyper)surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. In particular we are interested in rational solutions and present certain classes of equations having rational solutions. However, the method can also be used for finding non-rational solutions.

Symbolic Solutions of First-Order Algebraic ODEs

Algebraic ordinary differential equations are described by polynomial relations between the unknown function and its derivatives. There are no general solution methods available for such differential equations. However, if the hypersurface determined by the defining polynomial of an algebraic ordinary differential equation admits a parametrization, then solutions can be computed and the solvability in certain function classes may be decided. After an overview of methods developed in the last decade we present a new and rather general method for solving algebraic ordinary differential equations.

Radical solutions of first order autonomous algebraic ordinary differential equations

We present a procedure for solving autonomous algebraic ordinary differential equations (AODEs) of first order. This method covers the known case of rational solutions and depends crucially on the use of radical parametrizations for algebraic curves. We can prove that certain classes of AODEs permit a radical solution, which can be determined algorithmically. However, this approach is not limited to rational and radical solutions of AODEs.

Deciding the existence of rational general solutions for first-order algebraic ODEs

Abstract In this paper, we consider the class of first-order algebraic ordinary differential equations (AODEs), and study their rational general solutions. A rational general solution contains an arbitrary constant. We give a decision algorithm for finding a rational general solution, in which the arbitrary constant appears rationally, of the whole class of first-order AODEs. As a byproduct, this leads to an algorithm for determining a rational general solution of a class of first-order AODE which covers almost all first-order AODEs from Kamkes collection. The method is based intrinsically on the consideration of the AODE from a geometric point of view. In particular, parametrizations of algebraic curves play an important role for a transformation of a parametrizable first-order AODE to a quasi-linear differential equation.

On Symbolic Solutions of Algebraic Partial Differential Equations

In this paper we present a general procedure for solving first-order autonomous algebraic partial differential equations in two independent variables. The method uses proper rational parametrizations of algebraic surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. We will demonstrate in examples that, depending on certain steps in the procedure, rational, radical or even non-algebraic solutions can be found. Solutions computed by the procedure will depend on two arbitrary independent constants.

The Number of Realizations of a Laman Graph

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems.

Computing the number of realizations of a Laman graph

Abstract Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.

Lower Bounds on the Number of Realizations of Rigid Graphs

ABSTRACT Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Toward this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gröbner basis computations.

An Algebraic-Geometric Method for Computing Zolotarev Polynomials

In this paper we study a differential equation which arises from the theory of Zolotarev polynomials. By extending a symbolic algorithm for finding rational solutions of algebraic ordinary differential equations, we construct a method for computing explicit expressions for Zolotarev polynomials. This method is an algebraic geometric one and works subject to (radical) parametrization of algebraic curves. As a main application we compute the explicit form of the proper Zolotarev polynomial of degree 5.

Polynomial equivalence of finite rings

We prove that Zpn and Zp[t]/(t) are polynomially equivalent if and only if n ≤ 2 or p = 8. For the proof, employing Bernoulli numbers, we provide the polynomials which compute the carry-on part for the addition and multiplication in base p. As a corollary, we characterize nite rings of p elements up to polynomial equivalence.