Numerical computation of the homology of basic semialgebraic sets - PowerPoint PPT Presentation

Numerical computation of the homology of basic semialgebraic sets Pierre Lairez joint work with Peter Bürgisser and Felipe Cucker Inria Saclay TAGS 2018 Linking Topology to Algebraic Geometry and Statistics 22 February 2018, Leipzig

Basic semialgebraic sets definition A basic semi algebraic set is the solution set of finitely many polynomial equation and inequations. Picture : https://de.wikipedia.org/wiki/Steinmetz-Körper 1

Semialgebraic sets in applications motion planning J. T. Schwartz, M. Sharir, “On the piano movers problem” 2

Complexity bounds in real algebraic geometry (Koiran) CAD Compute the cylindrical algebraic decompositon (Collins) homology Compute the homology groups of double exponential algorithms — Euler Compute the Euler characteristic (Basu) Compute the first few Betti numbers (Basu) Grigoriev, Vorobjov) CC Compute the number of connected components (Canny, dimension Compute symbolic algorithms (Grigoriev, Vorobjov, Renegar) emptyness Decide if single exponential time algorithm — membership Decide if polynomial time algorithm . equations or inequalities of degree (basic) semialgebraic set defined by 3

Complexity bounds in real algebraic geometry CC Compute the number of connected components (Canny, CAD Compute the cylindrical algebraic decompositon (Collins) homology Compute the homology groups of double exponential algorithms — Euler Compute the Euler characteristic (Basu) Compute the first few Betti numbers (Basu) Grigoriev, Vorobjov) (Koiran) symbolic algorithms dimension Compute (Grigoriev, Vorobjov, Renegar) emptyness Decide if single exponential time algorithm — membership Decide if polynomial time algorithm 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D .

Complexity bounds in real algebraic geometry CC Compute the number of connected components (Canny, CAD Compute the cylindrical algebraic decompositon (Collins) homology Compute the homology groups of double exponential algorithms — Euler Compute the Euler characteristic (Basu) Compute the first few Betti numbers (Basu) Grigoriev, Vorobjov) (Koiran) symbolic algorithms dimension Compute (Grigoriev, Vorobjov, Renegar) emptyness Decide if single exponential time algorithm — membership Decide if polynomial time algorithm 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D .

Complexity bounds in real algebraic geometry CC Compute the number of connected components (Canny, CAD Compute the cylindrical algebraic decompositon (Collins) homology Compute the homology groups of double exponential algorithms — Euler Compute the Euler characteristic (Basu) Compute the first few Betti numbers (Basu) Grigoriev, Vorobjov) (Koiran) symbolic algorithms dimension Compute (Grigoriev, Vorobjov, Renegar) emptyness Decide if single exponential time algorithm — polynomial time algorithm 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W

Complexity bounds in real algebraic geometry CC Compute the number of connected components (Canny, CAD Compute the cylindrical algebraic decompositon (Collins) homology Compute the homology groups of double exponential algorithms — Euler Compute the Euler characteristic (Basu) Compute the first few Betti numbers (Basu) Grigoriev, Vorobjov) (Koiran) symbolic algorithms dimension Compute (Grigoriev, Vorobjov, Renegar) emptyness Decide if polynomial time algorithm 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1)

Complexity bounds in real algebraic geometry CC Compute the number of connected components (Canny, CAD Compute the cylindrical algebraic decompositon (Collins) homology Compute the homology groups of double exponential algorithms — Euler Compute the Euler characteristic (Basu) Compute the first few Betti numbers (Basu) Grigoriev, Vorobjov) (Koiran) symbolic algorithms dimension Compute polynomial time algorithm 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar)

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran)

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran) # CC Compute the number of connected components (Canny,

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm Grigoriev, Vorobjov) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran) # CC Compute the number of connected components (Canny, b 0 , b 1 , b 2 ,... Compute the first few Betti numbers (Basu)

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm Grigoriev, Vorobjov) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran) # CC Compute the number of connected components (Canny, b 0 , b 1 , b 2 ,... Compute the first few Betti numbers (Basu)

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm Grigoriev, Vorobjov) Euler Compute the Euler characteristic (Basu) homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran) # CC Compute the number of connected components (Canny, b 0 , b 1 , b 2 ,... Compute the first few Betti numbers (Basu) double exponential algorithms — ( sD ) 2 O ( n )

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm Grigoriev, Vorobjov) Euler Compute the Euler characteristic (Basu) CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran) # CC Compute the number of connected components (Canny, b 0 , b 1 , b 2 ,... Compute the first few Betti numbers (Basu) double exponential algorithms — ( sD ) 2 O ( n ) homology Compute the homology groups of W

Complexity bounds in real algebraic geometry symbolic algorithms polynomial time algorithm Grigoriev, Vorobjov) Euler Compute the Euler characteristic (Basu) CAD Compute the cylindrical algebraic decompositon (Collins) 3 W ⊆ R n (basic) semialgebraic set defined by s equations or inequalities of degree D . membership Decide if x ∈ W single exponential time algorithm — ( sD ) n O (1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dim W (Koiran) # CC Compute the number of connected components (Canny, b 0 , b 1 , b 2 ,... Compute the first few Betti numbers (Basu) double exponential algorithms — ( sD ) 2 O ( n ) homology Compute the homology groups of W

Approaching sets by union of balls numerical algorithms • Homotopically equivalent to its nerve • Combinatorial computation of the homology • Tricky choice of the parameters: • sufgiciently many points • radius not too small • radius not too large • How to quantify “sufgiciently many”, “too small” and “too large” in an algebraic setting? • Can we derive algebraic complexity bounds for the computation of the homology of semialgebraic sets? 4