Multi-degree Bounds on the Betti Numbers of Real Varieties and Semi-algebraic Sets and Applications

  title={Multi-degree Bounds on the Betti Numbers of Real Varieties and Semi-algebraic Sets and Applications},
  author={S. Basu and Anthony Rizzie},
  journal={Discrete \& Computational Geometry},
  • S. Basu, Anthony Rizzie
  • Published 14 July 2015
  • Mathematics, Computer Science
  • Discrete & Computational Geometry
We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially quadratic polynomials. The bounds we present in the case of partially quadratic polynomials offer a… Expand
Bounds on the individual Betti numbers of complex varieties, stability and algorithms
Graded bounds on the individual Betti numbers of affine and projective complex varieties are proved and certain homological and representational stability results for sequences of complex projective varieties which could be of independent interest are proved. Expand
Betti Numbers of Random Hypersurface Arrangements
We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in $\mathbb{R}\mathrm{P}^n$. Using a randomExpand
On generalizing Descartes' rule of signs to hypersurfaces
This approach opens a new route to generalize Descartes’ rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations. Expand
On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms
Let $$\mathrm {R}$$R be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$Rk defined byExpand
How to Flatten a Soccer Ball
This is an experimental case study in real algebraic geometry, aimed at computing the image of a semialgebraic subset of 3-space under a polynomial map into the plane. For general instances, theExpand
Topology of deep neural networks
The goal is to shed light on two mysteries in deep neural networks: (i) a nonsmooth activation function like ReLU outperforms a smooth one like hyperbolic tangent; (ii) successful neural network architectures rely on having many layers, even though a shallow network can approximate any function arbitrary well. Expand


Different bounds on the different Betti numbers of semi-algebraic sets
  • S. Basu
  • Computer Science, Mathematics
  • SCG '01
  • 2001
This paper proves separate bounds on the different Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. Expand
On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets
  • S. Basu
  • Mathematics, Computer Science
  • Discret. Comput. Geom.
  • 1999
It is proven that the sum of the Betti numbers of S is bounded by sk' 2O(k2 m4) in case the total number of monomials occurring in the polynomials in $ {\cal P} \cup \{Q\}$ is m. Expand
On the geometry of polar varieties
The aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Expand
On the Betti numbers of real varieties
PROOF. Approximate fi, * *, fm by real polynomials F1, * , Fm of the same degrees whose coefficients are algebraically independent. Now consider the variety Vc in the complex Cartesian spaceExpand
Polynomial Partitioning on Varieties of Codimension Two and Point-Hypersurface Incidences in Four Dimensions
A polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two is presented and a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space is presented. Expand
Deciding positivity of multisymmetric polynomials
This article generalizes the characterization of non-negative symmetric polynomials by adapting the method of proof developed in Riener (2013) and deduces that in the case of a fixed degree it is possible to derive a method to test for convexity which makes use of the special structure of (multi-)symmetric poynomials. Expand
Strong bi-homogeneous Bézout theorem and its use in effective real algebraic geometry
A bi-Homogeneous Bezout Theorem is proved, which bounds the sum of the degrees of the equidimensional components of the radical of an ideal generated by a bi-homogeneous polynomial family. Expand
On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets
  • S. Basu
  • Mathematics, Computer Science
  • STOC '96
  • 1996
New bounds are proved on the sum of the Betti numbers of closed semi-algebraic sets and the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi -al algebraic sets is given. Expand
Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique
This paper applies a new method for partitioning finite point sets in ℝd, based on the Stone–Tukey polynomial ham-sandwich theorem, to obtain new and simple proofs of two well known results: the Szemerédi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Expand
Bounding the Betti numbers and computing the Euler-Poincaré characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials
Let R be a real closed field, Q R(Y1;:::;Y';X1;:::;Xk); with degY.Q/ 2, degX.Q/ d, Q 2 Q, #.Q/ D m; and P R(X1;:::;Xk) with degX.P/ d, P 2 P, #.P/D s, and S R 'Ck a semi-algebraic set defined by aExpand