For more information about this meeting, contact Robert Vaughan.

Title: | Diagonal equations over finite fields |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Ivelisse Rubio, University of Puerto Rico |

Abstract: |

In general, it is difficult to find conditions that guarantee solvability of polynomial equations over finite fields. One of the best understood cases is the case of diagonal equations. For this case, there are very good bounds and estimates for the divisibility of the number of solutions are known. In this talk we will present some general aspects of diagonal equations over finite fields and some recent results on the Waring problem over finite fields. We will also present results on the solvability of families of systems of two polynomial equations over finite fields which are obtained by computing the exact divisibility of the exponential sums associated to the systems. This generalizes a theorem of Carlitz to systems of two equations. Our result gives
an upper bound for the Waring number of systems of diagonal equations. Also, as a by-product, we also obtain information about the p-divisibility of the number of solutions of the systems for cases for which the well known results of Chevalley-Warning and Katz do not give any information. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 09 / 01 / 2011 |

Time: | 11:15am - 12:05pm |