In this report, we present a number-theory-based approach for discrete tomography (DT),which is based on parallel projections of rational slopes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear equations with integer coefficients. Assuming that the range of pixel values is a(i, j) = 0, 1, . . . , M − 1, with M being a prime number, we reduce the equations modulo M ...

Introduction Something happened to me recently I would wager has happened to many who read this note. Teaching a new topic, you cannot understand one of the proofs. Your first attempt to fill the gap fails. You look through your books for an answer. Next, you ask colleagues, go to the library, maybe even use the interlibrary loan. All in vain. Then it strikes you that, in fact, you cannot answe...

What can be said about the set E ~ of solutions in nonnegative integers to a system of linear equations with integer coefficients? For many purposes, such as those of linear programming, this question has been adequately answered. However, when this question is regarded from the vantage point of commutative algebra, many additional aspects arise. In particular, there is a natural way to associa...

We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirical evidence obtained by directly comparing our methods with others, and which is partly presented here, we a...

In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly ind...

We present structural results on solutions to the Diophantine system Ay = b, y ∈ Z ≥0 with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel’s Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

Linear forms in logarithms have an important role the theory of Diophantine equations. In this paper, we prove explicit [Formula: see text]-adic lower bounds for linear rational numbers using Padé approximations second kind.