Farbod Shokrieh

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their

Improved simulation of nondeterministic Turing machines

{\mathbb {Z}}

Non-Archimedean and tropical theta functions

Z-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.

Chip-firing games, potential theory on graphs, and spanning trees

Abstract Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.

CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM

The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size S of this graph. This paper presents a new simulation method that runs in time O@?(S). The search savings exploit the one-dimensional nature of Turing machine tapes. In addition, we remove most of the time dependence on nondeterministic choices of states and tape head movements.

Divisors on Graphs, Connected Flags, and Syzygies

The standard simulation of a nondeterministic Turing machine (NTM) by a deterministic one essentially searches a large bounded-degree graph whose size is exponential in the running time of the NTM. The graph is the natural one defined by the configurations of the NTM. All methods in the literature have required time linear in the size S of this graph. This paper presents a new simulation method that runs in time O(√S). The search savings exploit the one-dimensional nature of Turing machine tapes. In addition, we remove most of the time-dependence on nondeterministic choice of states and tape head movements.

Chip-Firing Games,

We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud–Bosch–Lütkebohmert theory of non-Archimedean theta functions for abelian varieties with semi-abelian reduction.