Small points

Small Points and Adelic Metrics

Small Limit Points of Mahler's Measure

Let M(P (z1, . . . , zn)) denote Mahler’s measure of the polynomial P (z1, . . . , zn). Measures of polynomials in n variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer coefficients having small measure, demonstrate effective methods for computing these measures, and i...

Resolutions of small sets of fat points

We investigate the minimal graded free resolutions of ideals of at most n + 1 fat points in general position in Pn. Our main theorem is that these ideals are componentwise linear. This result yields a number of corollaries, including the multiplicity conjecture of Herzog, Huneke, and Srinivasan in this case. On the computational side, using an iterated mapping cone process, we compute formulas ...

Computing Points of Small Height for Cubic Polynomials

Let φ ∈ Q[z] be a polynomial of degree d at least two. The associated canonical height ĥφ is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of φ. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points, ĥφ is boun...

Global Discrepancy and Small Points on Elliptic Curves

Let E be an elliptic curve defined over a number field k. In this paper, we define the “global discrepancy” of a finite set Z ⊂ E(k) which in a precise sense measures how far the set is from being adelically equidistributed. We prove an upper bound for the global discrepancy of Z in terms of the average canonical height of points in Z. We deduce from this inequality a number of consequences. Fo...