An Algorithm to Compute the Topological Euler Characteristic, Chern-Schwartz-MacPherson Class and Segre Class of Projective Varieties

Let $V$ be a closed subscheme of a projective space $\mathbb{P}^n$. We give an algorithm to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of $V$. The algorithm can be implemented using either symbolic or numerical methods. The algorithm is based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Using this result and known formulas for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps we give algorithms to compute the Chern-Schwartz-MacPherson class and Segre class of a projective variety. Since the Euler characteristic of $V$ is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of $V$ our algorithm also computes the Euler characteristic $\chi(V)$. Relationships between the algorithm developed here and other existing algorithms are discussed. The algorithm is tested on several examples and performs favourably compared to current algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics

Nearest Points on Toric Varieties

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.Comment: 20 page

Segre Class Computation and Practical Applications

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used

Complexity of Model Testing for Dynamical Systems with Toric Steady States

In this paper we investigate the complexity of model selection and model testing for dynamical systems with toric steady states. Such systems frequently arise in the study of chemical reaction networks. We do this by formulating these tasks as a constrained optimization problem in Euclidean space. This optimization problem is known as a Euclidean distance problem; the complexity of solving this problem is measured by an invariant called the Euclidean distance (ED) degree. We determine closed-form expressions for the ED degree of the steady states of several families of chemical reaction networks with toric steady states and arbitrarily many reactions. To illustrate the utility of this work we show how the ED degree can be used as a tool for estimating the computational cost of solving the model testing and model selection problems

Polar degrees and closest points in codimension two

Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix $A$ in the codimension two case.Comment: 25 pages, 1 figur

Algorithms to Compute Characteristic Classes

In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases). We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and the Euler characteristic. The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Running time bounds are given for these algorithms and the algorithms are found to perform favourably compared to other applicable algorithms. Relations between our algorithms and other existing algorithms are explored. In the special case of a complete intersection subcheme we develop a second algorithm to compute CSM classes and Euler characteristics in a more direct and efficient manner. Each of these algorithms are generalized to subschemes of a product of projective spaces. Running time bounds for the generalized algorithms to compute the CSM class, Segre class and the Euler characteristic are given. Our Segre class algorithm is tested in comparison to another applicable algorithm and is found to perform favourably. To the best of our knowledge there are no other algorithms in the literature which compute the CSM class and Euler characteristic in the multi-projective setting. For complete simplical toric varieties defined by a fan we give a strictly combinatorial algorithm to compute the CSM class and Euler characteristic and a second combinatorial algorithm with reduced running time to compute only the Euler characteristic. We also prove several Bezout type bounds in multi-projective space. An application of these bounds to obtain a sharper degree bound on a certain system with a natural bi-projective structure is demonstrated