3 research outputs found
On the graph of a function over a prime field whose small powers have bounded degree
Let be a function from a finite field with a prime number of elements, to . In this article we consider those functions for which there is a positive integer with the property that , when considered as an element of , has degree at most , for all . We prove that every line is incident with at most points of the graph of , or at least points, where is a positive integer satisfying if is even and if is odd. With the additional hypothesis that there are lines that are incident with at least points of the graph of , we prove that the graph of is contained in these lines. We conjecture that the graph of is contained in an algebraic curve of degree and prove the conjecture for and . These results apply to functions that determine less than directions. In particular, the proof of the conjecture for and gives new proofs of the result of Lov\'asz and Schrijver \cite{LS1981} and the result in \cite{Gacs2003} respectively, which classify all functions which determine at most directions
On the graph of a function over a prime field whose small powers have bounded degree
Let be a function from a finite field with a prime number of elements, to . In this article we consider those functions for which there is a positive integer with the property that , when considered as an element of , has degree at most , for all . We prove that every line is incident with at most points of the graph of , or at least points, where is a positive integer satisfying if is even and if is odd. With the additional hypothesis that there are lines that are incident with at least points of the graph of , we prove that the graph of is contained in these lines. We conjecture that the graph of is contained in an algebraic curve of degree and prove the conjecture for and . These results apply to functions that determine less than directions. In particular, the proof of the conjecture for and gives new proofs of the result of Lov\'asz and Schrijver \cite{LS1981} and the result in \cite{Gacs2003} respectively, which classify all functions which determine at most directions
On the graph of a function over a prime field whose small powers have bounded degree
Let be a function from a finite field with a prime number of elements, to . In this article we consider those functions for which there is a positive integer with the property that , when considered as an element of , has degree at most , for all . We prove that every line is incident with at most points of the graph of , or at least points, where is a positive integer satisfying if is even and if is odd. With the additional hypothesis that there are lines that are incident with at least points of the graph of , we prove that the graph of is contained in these lines. We conjecture that the graph of is contained in an algebraic curve of degree and prove the conjecture for and . These results apply to functions that determine less than directions. In particular, the proof of the conjecture for and gives new proofs of the result of Lov\'asz and Schrijver \cite{LS1981} and the result in \cite{Gacs2003} respectively, which classify all functions which determine at most directions