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What is... Random Algebraic Geometry?

Antonio Lerario

Abstract

We survey some ideas from the subject of Random Algebraic Geometry, a field that introduces a probabilistic perspective on classical topics in real algebraic geometry. This offers a modern approach to classical problems, such as Hilbert's Sixteenth Problem.

What is... Random Algebraic Geometry?

Abstract

We survey some ideas from the subject of Random Algebraic Geometry, a field that introduces a probabilistic perspective on classical topics in real algebraic geometry. This offers a modern approach to classical problems, such as Hilbert's Sixteenth Problem.

Paper Structure

This paper contains 20 sections, 13 theorems, 104 equations, 2 figures.

Table of Contents

  1. Generic and random
  2. Complex and real discriminants
  3. From generic to random
  4. A little of Probability
  5. Gaussian random variables
  6. Random linear spaces
  7. A measure on the space of polynomials
  8. Degree and volume
  9. Integral geometry
  10. Newton polytopes
  11. The Veronese embedding and its geometry
  12. Probabilistic Intersection Theory
  13. Probabilistic Schubert Calculus
  14. Topology of random hypersurfaces
  15. The Betti numbers of a random algebraic hypersurface
  16. ...and 5 more sections

Key Result

Proposition 1

Let $P$ be a smooth manifold and $\Sigma\subset P$ be a closed submanifold--complex of codimension at least two. Then the inclusion $P\setminus \Sigma\hookrightarrow P$ induces an isomorphim on the zero--th homology.

Figures (2)

  • Figure 1: The discriminant in the space $P\simeq\mathbb{R}^3$ of real homogeneous polynomials of degree two is a quadric surface given by the equation $2a_{20}a_{02}-a_{11}^2=0$. Inside the cone we have $2a_{20}a_{02}>a_{11}^2$ and outside the cone $2a_{20}a_{02}<a_{11}^2$. Along a continuous path (the blue curve in the picture) between two polynomials, one inside the cone and the other outside the cone, we must hit $\Sigma$.
  • Figure 2: In the case $E=\{0,1\}\subset \mathbb{Z}$ the embedding $\nu_E:\mathbb{C}^*\to \mathbb{C}\mathrm{P}^1$ is given by $z\mapsto[1,z]$. In this case $T=\mathbb{C}\mathrm{P}^1$ and the moment map $\mu:\mathbb{C}\mathrm{P}^{1}\to \mathbb{R}$ is given by $[w_0, w_1]\mapsto\frac{|w_1|^2}{|w_0|^2+|w_1|^2}$. The image of $\mu$ is the segment $[0,1]=\mathrm{conv}(E).$

Theorems & Definitions (28)

  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Remark 3: The gaussian distribution in polar coordinates
  • Remark 4: Gaussian measures and scalar products
  • Remark 5
  • Theorem 6: The Integral Geometry Formula for $\mathbb{R}\mathrm{P}^n$
  • Theorem 7: Bernstein--Khovanskii--Kushnirenko
  • Theorem 8
  • ...and 18 more