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Washington State
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Combinatorics, Linear Algebra and Number Theory Seminar

Department of Mathematics and Statistics

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April 11, Monday, 4:10 - 5:00 PM

Derek Garton

Fariborz Maseeh Department of Mathematics and Statistics

Portland State University

Combinatorics, Linear Algebra and Number Theory Seminar

Department of Mathematics and Statistics

Zoom

April 11, Monday, 4:10 - 5:00 PM

Derek Garton

Fariborz Maseeh Department of Mathematics and Statistics

Portland State University

Title: Periodic
points of polynomials over finite fields

Abstract: A discrete dynamical system consists of a set and a function from that set to itself. As an example, if we let our set be F_7 (a finite field of size 7), then there are 7^7=823543 such functions. On the other hand, if we consider only those functions induced by quadratic polynomials (with coefficients in F_7), then we have restricted the size of our pool of functions to a mere 7^3-7^2=294.

In this talk, we investigate the extent to which the statistics of dynamical systems induced by quadratic polynomials over finite fields match those of random dynamical systems on finite sets. Our focus will be the particular statistic of ``periodic proportion". Given a dynamical system consisting of a finite set S and a function f, the periodic proportion of the system is the proportion of S that is periodic with respect to f. We will show that the average periodic proportion of quadratic dynamical systems tends to zero as the size of the finite field increases, just as it does for random dynamical systems. The proof of this result incorporates tools from algebraic geometry, Galois theory, and height functions on global fields, and we will describe how these tools work together to yield this result.

Abstract: A discrete dynamical system consists of a set and a function from that set to itself. As an example, if we let our set be F_7 (a finite field of size 7), then there are 7^7=823543 such functions. On the other hand, if we consider only those functions induced by quadratic polynomials (with coefficients in F_7), then we have restricted the size of our pool of functions to a mere 7^3-7^2=294.

In this talk, we investigate the extent to which the statistics of dynamical systems induced by quadratic polynomials over finite fields match those of random dynamical systems on finite sets. Our focus will be the particular statistic of ``periodic proportion". Given a dynamical system consisting of a finite set S and a function f, the periodic proportion of the system is the proportion of S that is periodic with respect to f. We will show that the average periodic proportion of quadratic dynamical systems tends to zero as the size of the finite field increases, just as it does for random dynamical systems. The proof of this result incorporates tools from algebraic geometry, Galois theory, and height functions on global fields, and we will describe how these tools work together to yield this result.