Past Junior Number Theory Seminar

8 February 2010
16:00
Sebastian Pancratz
Abstract
Let $X$ be a smooth hypersurface in projective space over a field $K$ of characteristic zero and let $U$ denote the open complement. Then the elements of the algebraic de Rham cohomology group $H_{dR}^n(U/K)$ can be represented by $n$-forms of the form $Q \Omega / P^k$ for homogeneous polynomials $Q$ and integer pole orders $k$, where $\Omega$ is some fixed $n$-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.
  • Junior Number Theory Seminar
1 February 2010
16:00
Abstract
Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a $p$-adic field $k$. A special case of a conjecture of Artin is that the forms $C$ and $C'$ have a common zero over $k$. While the conjecture of Artin is false in general, we try to argue that, in this case, it is (almost) correct! This is still work in progress (joint with Heath-Brown), so do not expect a full answer. As a historical note, some cases of Artin's conjecture for certain hypersurfaces are known. Moreover, Jahan analyzed the case of the simultaneous vanishing of a cubic and a quadratic form. The approach we follow is closely based on Jahan's approach, thus there might be some overlap between his talk and this one. My talk will anyway be self-contained, so I will repeat everything that I need that might have already been said in Jahan's talk.
  • Junior Number Theory Seminar
18 January 2010
16:00
Timothy Trudgian
Abstract
<p> How many integer-points lie in a circle of radius $\sqrt{x}$? </p> <p> A poor man's approximation might be $\pi x$, and indeed, the aim-of-the-game is to estimate </p> <p> $$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$ </p> <p> Once one gets the eye in to show that $P(x) = O(x^{1/2})$, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $P(x) = O(x^{\alpha})$ can only possible hold when $\alpha \geq 1/4$ there is some room for improvement in the middle-order. </p> <p> In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $P(x)$. </p>
  • Junior Number Theory Seminar

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