On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree

@article{Ishitsuka2019OnAT,
  title={On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree},
  author={Yasuhiro Ishitsuka and Tetsushi Ito and T. Ohshita},
  journal={JSIAM Lett.},
  year={2019},
  volume={11},
  pages={9-12}
}
We give two algorithms to compute linear determinantal representations of smooth plane curves of any degree over any field. As particular examples, we explicitly give representatives of all equivalence classes of linear determinantal representations of two special quartics over the field $\mathbb{Q}$ of rational numbers, the Klein quartic and the Fermat quartic. This paper is a summary of third author's talk at the JSIAM JANT workshop on algorithmic number theory in March 2018. Details will… Expand
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References

SHOWING 1-10 OF 15 REFERENCES
An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields
TLDR
This paper is a summary of the author's talk at the JSIAM JANT workshop on algorithmic number theory in March, 2016, and gives an algorithm to obtain all linear determinantal representations up to equivalence. Expand
On the symmetric determinantal representations of the Fermat curves of prime degree
We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rationalExpand
Quartic curves and their bitangents
TLDR
Interwoven is an exposition of much of the 19th century theory of plane quartics, which expresses Vinnikov quartics as spectrahedra and positive Quartics as Gram matrices and explores the geometry of Gram spectahedra. Expand
The local–global principle for symmetric determinantal representations of smooth plane curves
A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We studyExpand
Low-degree points on Hurwitz-Klein curves
We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effectiveExpand
Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic
We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsionExpand
Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve
We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of theExpand
A positive proportion of cubic curves over Q admit linear determinantal representations
Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. InExpand
Classical Algebraic Geometry: A Modern View
Algebraic geometry has benefited enormously from the powerful general machinery developed in the latter half of the twentieth century. The cost has been that much of the research of previousExpand
Determinantal hypersurfaces
Introduction (0.1) We discuss in this paper which homogeneous form on P n can be written as the determinant of a matrix with homogeneous entries (possibly symmetric), or the pfaaan of aExpand
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