Probabilistic Saturations and Alt's Problem

08/15/2019
by   Jonathan D. Hauenstein, et al.
0

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt's problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/13/2023

Fast evaluation and root finding for polynomials with floating-point coefficients

Evaluating or finding the roots of a polynomial f(z) = f_0 + ⋯ + f_d z^d...
research
12/20/2021

The complexity of solving Weil restriction systems

The solving degree of a system of multivariate polynomial equations prov...
research
07/04/2021

Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms

We analyse the forward error in the floating point summation of real num...
research
02/12/2018

Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations

Floating point error is a drawback of embedded systems implementation th...
research
08/04/2023

Kernelization of Counting Problems

We introduce a new framework for the analysis of preprocessing routines ...
research
03/14/2020

Intersection distribution, non-hitting index and Kakeya sets in affine planes

We propose the concepts of intersection distribution and non-hitting ind...
research
12/15/2017

Counting Solutions of a Polynomial System Locally and Exactly

We propose a symbolic-numeric algorithm to count the number of solutions...

Please sign up or login with your details

Forgot password? Click here to reset