Related papers: A modern solution to the Gion shrine problem
In this work we present a new polynomial map $f:=(f_1,f_2):{\mathbb R}^2\to{\mathbb R}^2$ whose image is the open quadrant $\{x>0,y>0\}\subset{\mathbb R}^2$. The proof of this fact involves arguments of topological nature that avoid hard…
In this paper we present a unified method for solving general polynomial equations of degree less than five.
A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance…
These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…
In this paper, we give a survey of a geometrical theory of Jacobi forms of higher degree. And we present some geometric results and discuss some geometric problems to be investigated in the future.
As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic…
The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time…
A possible method to solve the sign problem is developed by modifying the original theory. Considering several modifications of the partition function, the observable in the original theory is reconstructed from the identity connecting the…
The aim of this paper is to generalize Apollonius' problem. The problem is to construct a circle that is tangent to three given circles in a plane. We find the maximum possible number of solution circles in the case of more than the three…
We propose an algorithm for solving of the graph isomorphism problem. Also, we introduce the new class of graphs for which the graph isomorphism problem can be solved polynomially using the algorithm.
Illumination of scenes is usually generated in computer graphics using polygonal meshes. In this paper, we present a geometric method using projections. Starting from an implicit polynomial equation of a surface in 3-D or a curve in 2-D, we…
Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of geometric primitives, e.g., splines, triangles, and hexahedra,…
Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.
We present a concise self-contained inversive geometry solution of the three-circle problem of Steiner of constructing a circle that intersects each of the three given circles at one of the three given angles.
The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.
The Generalized Discretizable Molecular Distance Geometry Problem is a distance geometry problems that can be solved by a combinatorial algorithm called ``Branch-and-Prune''. It was observed empirically that the number of solutions of YES…
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these…
We consider quantum algorithms for the unique sink orientation problem on cubes. This problem is widely considered to be of intermediate computational complexity. This is because there no known polynomial algorithm (classical or quantum)…
In this paper we give a complete solution of the following "polynomial moment problem" which arose about 10 years ago in connection with Poincare's center-focus problem. For a given polynomial P(z) to describe polynomials Q(z) orthogonal to…
We study the domain of existence of a solution to a Riemann problem for the pressure gradient equation in two space dimensions. The Riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum…