Real tropical hyperfaces by patchworking in polymake Michael Joswig - PowerPoint PPT Presentation

Real tropical hyperfaces by patchworking in polymake Michael Joswig TU Berlin & MPI-MiS Braunschweig/online, 14 July 2020 joint w/ Paul Vater (MPI-MiS)

Real Tropical Geometry For a real parameter 0 < t ≪ 1 consider, e.g., + x 3 − tx 2 y + tx 2 z − t 4 xy 2 − t 3 xyz − t 4 xz 2 − t 9 y 3 + t 7 y 2 z − t 7 yz 2 − t 9 z 3 . Taking the limit lim t → 0 log t ( · ) yields the tropical polynomial � min 3 X , 1+2 X + Y , 1+2 X + Z , 4+ X +2 Y , 3+ X + Y + Z , � 4+ X +2 Z , 9+3 Y , 7+2 Y + Z , 7+ Y +2 Z , 9+3 Z . It vanishes at ( X , Y , Z ) if that minimum is attained at least twice. • methods of polyhedral geometry apply • Viro’s patchworking: take signs into account to describe the real locus Here: new implementation of combinatorial patchworking in polymake Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020

Example: Harnack Curve of Degree d = 3 0 1 0 1 1 1 0 1 1 1 • Harnack (1876): The number of connected components of a plane projective real algebraic curve of degree d is at most 1 2( d − 1)( d − 2) + 1 . Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020

Hilbert’s 16th Problem Task: Classify the isotopy types of real algebraic curves and surfaces. • Viro (1979): curves of degree d ≤ 7 • Kharlamov (1972): surfaces w/ d ≤ 4 • Viro (1980) / Itenberg (1993): counter-examples to Ragsdale’s conjecture via patchworking • Renaudineau & Shaw (2018) G¨ ott. Nachr., Heft 3, 253–297 (1900) Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020

A Census of Betti Numbers of Patchworked Surfaces • d = 3: 1 000 000 random triangulations w/ 20 random sign vectors • d = 4: 100 000 random triangulations w/ 20 random sign vectors • d ∈ { 5 , 6 } : already more difficult to generate many samples 10 6 b 0 = 1 b 0 = 1 d = 4 10 7 b 0 = 2 b 0 = 2 10 5 d = 3 10 6 10 4 10 3 10 5 10 2 10 4 10 1 10 3 10 0 1 3 5 7 0 2 4 6 8 10 12 14 16 18 20 • Jordan, J. & Kastner (2018): enumeration of all 21 125 102 (orbits of) regular and full triangulations of 3 · ∆ 3 [ MPTOPCOM ] Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020

Implementation in polymake 10 4 Viro.sage 10 4 polymake 10 3 10 3 10 2 10 2 10 1 10 1 10 0 10 0 10 − 1 Viro.sage polymake 10 − 1 0 20 40 60 80 100 3 4 5 6 curves by degree surfaces by degree • Input: tropical polynomial & sign vector • compute regular triangulation of support • construct chain complexes with Z 2 coefficients • Output: Z 2 Betti numbers via Gauß elimination Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020

Conclusion • polymake implementation of Viro’s combinatorial patchworking • fast enough to compute many examples • supports non-regular patchworking real tropical cubic surface with b 0 = 2 www.polymake.org Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020