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Summer School on Numerical Computing in Algebraic Geometry - Why attend?

Emre Sertöz

Numerical methods to solve polynomial systems arose to address problems in industry and engineering [1], and as a result developed an emphasis on efficiency and experimentation over rigor. However, in the past two decades there has been growing interest in applying these numerical tools to algebraic geometry [2,3]. During this time, certifications for numerical methods were developed and implemented [4,5,6] in order to give rigorous proofs.
Below, we will describe types of problems in algebraic geometry where numerical methods are put to good use. If one of them appeals to you, you might be interested in the conference. Needless to say, the list is not exhaustive.

Enumerative geometry

Solving a polynomial system numerically and with certification reveals, in particular, the number of solutions of that system. This is akin to enumerative geometry, which intends to count the number of solutions satisfied by a geometric problem that admits finitely many solutions. Typically, enumerative geometry is most useful when the problem is too difficult to solve explicitly. However, immense polynomial systems can now be solved by numerical methods, for instance by Bertini [7] or HomotopyContinuation.jl [8]. Therefore, rephrasing an enumerative problem in terms of explicit equations can lead to its solution by brute force.

Primary decomposition

One can also use numerical methods to study geometric properties of high dimensional varieties, e.g., the number of irreducible components of a projective variety, along with the dimension and degree of each component [7]. This information is often very expensive to obtain by relying purely on symbolic methods, for instance, by computing the prime decomposition of an ideal.

Galois group of a cover

One can compute elements of the Galois group of a cover X -> Y by lifting loops in Y to paths in X. The lifting operation can be performed numerically and with certification [5]. Combined with symbolic methods in group algebra as well as a theoretical study of the problem, one can completely determine the monodromy group of a finite covering, see [9] for a demonstration.

Computing cohomology and Hilbert polynomials of ideal sheaves

Besides solving polynomials, numerical methods have already been successfully applied in other subjects, most notably in linear algebra. A standard tool in numerical linear algebra is the singular value decomposition of a matrix, which computes the "probable" rank of an approximately given matrix. This method is employed in algebraic geometry to compute Hilbert polynomials of projective varieties. In fact, a slight generalization allows for the computation of the dimension of the cohomologies of an ideal sheaf [10].

K3 surfaces with real multiplication

Elsenhans and Jahnel [11] compute the equations of a curve C in the moduli space of K3 surfaces of Picard rank 16, such that the generic K3 parametrized by this curve has a particular type of real multiplication. This uses predictor-corrector methods employed on the period map to sample points on C. Then a singular value decomposition method is used to guess the equations of the curve from these points. This illustrates the subtle arithmetic questions one can study with an insightful use of numerical methods.

Decomposing Jacobians of curves

In [12] Jacobians of curves are constructed explicitly using a numerical computation of periods of curves. The endomorphism rings of these Jacobians are then determined using lattice basis reduction methods (LLL). From these endomorphism rings, one can determine if the Jacobian decomposes. As a result, structural properties of the curve can be discovered, such as the automorphism groups, and thus the non-trivial quotients, of the curve.

References

  1. A.J Sommese and C.W. Wampler. The Numerical solution of systems of polynomials arising in engineering and science. World Scientific Press, Singapore, 2005
  2. D.J Bates, B. GianMario, S.D. Rocco and C.W. Wampler (eds). Interactions of Classical and Numerical Algebraic Geometry. Contemporary Mathematics. Volume 496. American Mathematical Society. Providence, Rhode Island. 2008.
  3. A.J. Sommese and C.W. Wampler. Numerical algebraic geometry. The Mathematics of Numerical Analysis: Real Number Algorithms. Park City, Utah. 1995.
  4. J.D. Hauenstein and F. Sottile, Algorithm 921: alphaCertified: Certifying solutions to polynomial systems, ACM Trans. Math. Softw., 38 (2012), p. 28.
  5. J.D. Hauenstein and Alan C. Liddell, Jr., Certified predictor-corrector tracking for Newton homotopies. Journal of Symbolic Computation. Volume 74. 2016.
  6. M. Shub and S. Smale, On the average cost of solving polynomial equations. Geometric dynamics. Geometric dynamics.Lecture Notes in Math. Springer, Berlin. 1983.
  7. D.J. Bates, J.D. Hauenstein, and A.J. Sommese and C.W. Wampler. Numerically solving polynomial systems with Bertini. Software, Environments, and Tools. Volume 25. SIAM, Philadelphia. 2013.
  8. P. Breiding and T. Sascha. HomotopyContinuation.jl - a package for solving systems of polynomial equations in Julia. arXiv:1711.10911.
  9. J.D. Hauenstein, J. Rodriguez and B. Sturmfels. Maximum Likelihood for Matrices with Rank Constraints. arXiv:1210.0198.
  10. J.D. Hauenstein, J.C. Migliore, C. Peterson, and A.J. Sommese. Numerical computation of the dimensions of the cohomology of twists of ideal sheaves. Contemporary Mathematics, 496, 235-242, 2009.
  11. A.S. Elsenhans and J.Jahnel. Real and complex multiplication on K3 surfaces via period integration. arXiv:1802.10210.
  12. N. Bruin, J. Sijsling and A. Zotine. Numerical computation of endomorphism rings of Jacobians. Submitted.