Extensions of Hilbert's Tenth Problem and Computational Aspects of Arithmetic Geometry

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The investigator will work on two projects connected with numbertheory and arithmetic geometry. The first project is to studygeneralizations of Hilbert's Tenth Problem. The problem in itsoriginal form asked for an algorithm to decide whether an arbitrarymultivariable polynomial equation with integer coefficients has aninteger solution. In 1970 Matiyasevich proved that no such algorithmexists, i.e. Hilbert's Tenth Problem is undecidable. This motivatedstudying analogues of this problem by considering equations andsolutions in other commutative rings. The biggest open problem in thearea is Hilbert's Tenth Problem over the rational numbers. The PI hasproved the undecidability of Hilbert's Tenth Problem for variousfunction fields. These generalizations have used tools fromarithmetic geometry, such as the study of rational points on ellipticcurves. One research goal is to extend these results and proveundecidability of Hilbert's Tenth Problem for the function fields for whichthe problem is still unresolved. The biggest open problems arefunction fields of one variable over an algebraically closed field.Another goal is to explore Hilbert's Tenth Problem for varioussubrings of number fields. The second project is to study severalproblems that deal with computational aspects of curves and theirJacobians. Elliptic curves and, more generally, Jacobians of curvesof small genus have many applications to cryptography, and the secondproject focuses on these applications. One goal of the second projectis to explore curves of small genus and work on constructing curvesthat are suitable for cryptographic purposes. The PI will also work onapplications of pairings to cryptography.Both projects involve studying the solutions to multivariablepolynomial equations. Looking for solutions to such equations over theintegers or rational numbers has a long history that goes back toancient Greece. For the first project the investigator will study thefundamental question of whether it is possible to find a procedurethat determines whether an arbitrary multivariable polynomial equationhas a solution in a given number system. The second project focuseson computational aspects of certain special classes of equations thathave applications to cryptography. For these applications one usuallylooks for solutions to these equations over finite fields.