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., +x3−tx2y+tx2z−t4xy2−t3xyz−t4xz2−t9y3+t7y2z−t7yz2−t9z3 . Taking the limit limt→0 logt(·) yields the tropical polynomial min

• 3X, 1+2X+Y , 1+2X+Z, 4+X+2Y , 3+X+Y +Z,

4+X+2Z, 9+3Y , 7+2Y +Z, 7+Y +2Z, 9+3Z

• .

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

1 1 1 1 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.

G¨

• tt. Nachr., Heft 3, 253–297 (1900)
• 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)

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

1 3 5 7 103 104 105 106 107 d = 3 b0 = 1 b0 = 2 10 12 14 16 18 2 20 4 6 8 100 101 102 103 104 105 106 d = 4 b0 = 1 b0 = 2

• 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

20 40 60 80 100 10−1 100 101 102 103 104 Viro.sage polymake

curves by degree

3 4 5 6 10−1 100 101 102 103 104 Viro.sage polymake

surfaces by degree

• Input: tropical polynomial & sign vector
• compute regular triangulation of support
• construct chain complexes with Z2 coefficients
• Output: Z2 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 b0 = 2

www.polymake.org

Michael Joswig (TU Berlin & MPI-MiS) Patchworking in polymake 14 July 2020