• Corpus ID: 116432348

Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements

  title={Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements},
  author={Marco Pizzato},
In this thesis we consider some problems concerning polynomials over finite fields. The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers. Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained. Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for… 
4 Citations
Generalizations of self-reciprocal polynomials
Irreducible polynomials from a cubic transformation
Let R(x) = g(x)/h(x) be a rational expression of degree three over the finite field Fq. We count the irreducible polynomials in Fq[x], of a given degree, which have the form h(x) f · f ( R(x) ) for
Cubic rational expressions over a finite field
We classify the cubic rational expressions g(x)/h(x) over a finite field, having at most three ramification points, under an equivalence relation given by preand post-composition with independent
Birational Maps in the Minimal Model Program.
In this dissertation I face three main arguments. 1) Classification of Fano-Mori contractions. 2) Chern numbers on smooth threefolds. 3) Pluricanonical systems.


Further results on a class of permutation polynomials over finite fields
Rational functions with given ramification in characteristic p
Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of rational functions (up to automorphism of the target) on
On the construction of irreducible self-reciprocal polynomials over finite fields
  • H. Meyn
  • Mathematics
    Applicable Algebra in Engineering, Communication and Computing
  • 2005
Infinite sequences of irreducible self-reciprocal polynomials are constructed by iteration of thisQ-transformation.
Generalization of a theorem of Carlitz
  • O. Ahmadi
  • Mathematics
    Finite Fields Their Appl.
  • 2011
The arithmetic of elliptic curves
  • J. Silverman
  • Mathematics, Computer Science
    Graduate texts in mathematics
  • 1986
It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
We give an explicit description of the 𝔽qi-rational points on the Fermat curve uq-1 + vq-1 + wq-1 = 0, for i ∈{1, 2, 3}. As a consequence, we observe that for any such point (u, v, w), the product
Planar functions over finite fields
The main result is that every planar function is a quadratic polynomial and the following characterization of desarguesian planes of prime order is derived.