Dataset Open Access

Certain rigid maximally mutable Laurent polynomials in three variables

Coates, Tom; Kasprzyk, Alexander; Pitton, Giuseppe

This dataset contains certain rigid maximally mutable Laurent polynomials (rigid MMLPs) in three variables. Rigid MMLPs are defined in reference [1].

The Newton polytopes of these Laurent polynomials are three-dimensional canonical Fano polytopes. That is, they are three-dimensional convex polytopes with vertices that are primitive integer vectors and that contain exactly one lattice point, the origin, in their strict interior. See references [2] and [3].

Although the rigid MMLPs specified in this dataset have 3-dimensional canonical Fano Newton polytope, this is by no means an exhaustive list of such Laurent polynomials. The dataset contains examples of rigid MMLPs that correspond under mirror symmetry to three-dimensional Q-Fano varieties of particulaly high estimated codimension: see reference [4].

The file "rigid_MMLPs.txt" contains key:value records with keys and values as described below, separated by blank lines. Each key:value record determines a rigid MMLP, and there are 130 records in the file. An example record is:

canonical3_id: 231730
coefficients: [1,1,1,1,1,1,1,1,1,1]
exponents: [[-1,-1,-1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,2,2],[2,1,2],[2,1,3],[3,3,5],[4,2,5]]
period: [1,0,0,12,24,0,540,2940,2520,33600,327600,693000,2795100,35315280,129909780,354666312,3816572760,20559258720,59957561664,435508321248,2969362219824]
ulid: 01G5CBH3F86NRYF0TJ8MYWM41H

The keys and values are as follows, where f denotes the Laurent polynomial defined by the key:value record.

canonical3_id: an integer, the ID of the Newton polytope of f in reference [3]
coefficients: a string of the form "[c1,c2,...,cN]" where c1, c2, ... are integers. These are the coefficients of f.
exponents: a string of the form "[[x1,y1,z1],[x2,y2,z2],...,[xN,yN,zN]]" where x1, y1, z1, ..., xN, yN, zN are integers. These are the exponents of f.
period: a string of the form "[d0,d1,...,d20]" where d0, d1, ..., d20 are non-negative integers that give the first 21 terms of the period sequence for f.
ulid: a string that uniquely identified this entry in the dataset

The sequences defined by the keys "coefficients" and "exponents" are parallel to each other. The period sequence for f is defined, for example, in equations 1.2 and 1.3 of reference [1].


[1] Tom Coates, Alexander M. Kasprzyk, Giuseppe Pitton, and Ketil Tveiten. Maximally mutable Laurent polynomials. Proceedings of the Royal Society A 477, no. 2254:20210584, 2021.

[2] Alexander M. Kasprzyk. Canonical toric Fano threefolds. Canadian Journal of Mathematics, 62(6):1293–1309, 2010.

[3] Alexander M. Kasprzyk. The classification of toric canonical Fano 3-folds. Zenodo,, 2010.

[4] Liana Heuberger. Q-Fano threefolds and Laurent inversion. Preprint, arXiv:2202.04184, 2022.

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