Diophantine equations in two variables by Kim, M

The solubility of diagonal cubic diophantine equations by Heath-Brown, D

Ω in number theory by Ord, T, Kieu, TD

“…We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be used to create polynomials which express the bits of Ω in the number of positive values they assume.…”

On relative computability for curves by Kim, M

“…We discuss a rational version of a conjecture of Matiyasevich, Davis, and Putnam on the relative decidability of the finiteness problem for Diophantine equations with respect to the existence problem. …”

Semidefinite programming and linear equations vs. homomorphism problems by Ciardo, L, Zivny, S

“…We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. …”

On the existential theories of Büchi arithmetic and linear p-adic fields by Guépin, F, Haase, C, Worrell, J

“…A key technical contribution is to show that the existence of a path between two states of a finite-state automaton whose language encodes the set of solutions of a given system of linear Diophantine equations can be witnessed in NP.…”

Counting string theory standard models by Constantin, A, He, Y-H, Lukas, A

“…This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kähler parameters. …”

Costs and rewards in priced timed automata by Fränzle, M, Shirmohammadi, M, Swaminathan, M, Worrell, J

“…</p> <p>We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10th Problem, while decidability for three observers entails translation to a decidable fragment of arithmetic involving quadratic forms.…”

Costs and rewards in priced timed automata by Fränzle, M, Shirmohammadi, M, Swaminathan, M, Worrell, J

“…We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10thProblem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms.…”

Topics in sieve theory by McGrath, O

“…</p> <p>The second problem asks us to show that a certain Diophantine equation has few integer solutions. We approach this problem by adapting the polynomial sieve developed by Browning and using the work of Weil and Deligne to obtain appropriate estimates for various exponential sums which naturally arise. …”

Heterotic string compactification and quiver gauge theory on toric geometry by Sun, C

“…After explicitly computing the Diophantine equation of five block cases, we use this structure to re-organize the result in a form that can be applied to arbitrary block numbers. …”