Graded Syzygies: Geometry and Computation

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This research focuses on several problems in computational commutative algebra and algebraic geometry. This project is concerned with the study of systems of polynomial equations in several variables. Polynomial equations model many real world phenomena, ranging from robotics motion planning to physics to conservation programs. To each system of polynomial equations, we associate a set of points, called a variety, which correspond to the set of solutions for all the equations. There is a growing body of literature that shows that the computational complexity of the system of polynomial equations has deep connections to the geometry of the associated variety. In nice cases, when the associated variety is smooth, there are well understood bounds limiting the computational complexity. In the worst possible cases, complexity is doubly exponential. Recent work of the PI shows there is a middle ground which is still poorly understood. The goal of this research is to better understand this connection.The research aims to attack several open questions concerning projective bounds on syzygies. One goal is to study Rees-Like Algebras, which were essential in the PI's construction of counterexamples to the Eisenbud-Goto Conjecture (joint with I. Peeva), and to relate their properties to the more well-studied Rees Algebras. A second goal is to provide effective bounds on invariants, such as the Castelnuovo-Mumford regularity or the degrees of individual syzygies, in terms of other invariants. All of the projects have a computational flavor and include writing code for Macaulay2, an NSF-sponsored computer algebra system maintained by Grayson and Stillman. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.