BCC, day 5

My route to the conference passes a stagnant ditch. Today, two families of moorhens were in evidence; a couple with two small chicks, and another group of three older chicks. Blackbirds are also busy foraging on the lawn outside the window at breakfast.

The day began with a really beautiful talk by Gary McGuire, who is (among other things) the person responsible for showing that there are no 16-clue Sudokus. He talked about plane curves over finite fields and the connection with linear recurrent sequences. There was a lot of algebraic geometry just below the surface, and the names of Weil and Serre, Jacobians, and Picard groups were invoked from time to time; but everything was remarkably clear. His computations of zeta-functions of plane curves have revealed some amazing patterns, some of which can be proved. The zeta-function of a plane curve over the field of q elements has the form L(t)/(1−t)(1−qt), and he is interested in when the L-polynomial of one curve divides that of another. If the quotient is a polynomial in tk, then the numbers of points on the two curves are equal over extensions with degree not divisible by k. Since the generating function for the difference between the number of points over an extension of degree n and the “expected” number 1+qn is rational, this series satisfies a linear recurrence, which has stronger properties than are immediately apparent.

After coffee, my freedom was restricted by the fact that I had to chair a session, but the talks in the session were all things I wanted to hear anyway. David Penman counted linear extensions of posets, being interested in the maximum and minimum numbers possible for given numbers of comparabilities in the poset; Joanna Fawcett described locally triangular graphs with a sufficient amount of symmetry; Mark Ellingham was extending Whitney’s results on line graphs to “link graphs”, and needed ideas from topological graph theory to do it; and Anurag Bishnoi presented a remarkably general technique relating putting balls in boxes to geometric problems of Chevalley–Warning type including Jamison’s result on blocking sets in the affine plane.

We had lunch, and then went our separate ways.


About Peter Cameron

I count all the things that need to be counted.
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One Response to BCC, day 5

  1. Here’s another way of looking at Chevalley-Warning theorem (along with its restricted variable generalisation by Brink/Schauz/Wilson) and Jamison’s result in a uniform manner: https://anuragbishnoi.wordpress.com/2015/05/26/chevalley-warning-theorem-and-blocking-sets/. This is essentially the Brouwer-Schrijver proof written down in a way that the property of polynomials vanishing on all points except one is made explicit.

    This property was generalised by Simeon Ball and Oriol Serra in their paper titled “Punctured Combinatorial Nullstellensatz” where they show that a polynomial which vanishes everywhere on a grid A = A1 x … x An except at some point of a subgrid B = B1 x …. x Bn, then its degree is at least \sum (|Ai| – |Bi|). From this I derived a new generalisation of the Chevalley-Warning theorem which doesn’t seem to follow from the Alon-Furedi bound (hence I didn’t include it in my talk). Nevertheless, it seems to be quite interesting on its own.

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