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Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (\emph{absolutely irreducibles}) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials $F \in \operatorname{Int}(R)$ where $R$ is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number $S\in \mathbb{N}$ that reduces the absolute irreducibility of $F$ to the unique factorization of $F^S$. To this end, we establish a connection between the factors of powers of $F$ and the kernel of a certain linear map that we associate to $F$. This connection yields a characterization of absolute irreducibility in terms of this so-called \emph{fixed divisor kernel}. Given a non-trivial element $\boldsymbol{v}$ of this kernel, we explicitly construct non-trivial factorizations of $F^k$, provided that $k\ge L$, where $L$ depends on $F$ as well as the choice of $\boldsymbol{v}$. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for $k$, one of which only depends on the valuation of the denominator of $F$ and the size of the residue class field of $R$.

International audience; We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.

It has been an open question whether any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. A positive proposal is also known as the “Ize-conjecture”. Algebraically speaking, this is to ask whether every absolutely irreducible real representation has an odd dimensional fixed point subspace corresponding to some subgroups. Recently, Reiner Lauterbach and Paul Matthews have found counter examples to this conjecture and interestingly, all of the representations are of dimension $4k$, for $k\in\mathbb{N}$. A natural question arises: what about the case $4k+2$? ¶ In this paper, we give a partial answer to this question and prove that in any $6$-dimensional absolutely irreducible real representation of a finite solvable group, there exists an odd dimensional fixed point subspace with respect to subgroups.

In his volume [5] on "Symmetry Breaking for Compact Lie Groups" Mike Field quotes a private communication by Jorge Ize claiming that any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. The proof should come from the fact that any absolutely irreducible representation possesses an odd dimensional fixed point space. In this paper we show that there are many examples of groups which have absolutely irreducible representations but no odd dimensional fixed point space. This observation may be relevant also for some degree theoretic considerations concerning equivariant bifurcation. Moreover we show that our examples give rise to some interesting Hamiltonian dynamics and we show that despite some complications we can go a long way towards doing explicit computations and providing complete proofs. For some of the invariant theory needed we will depend on some computer aided computations. The work presented here greatly benefited from the computer algebra program GAP [6], which is an indispensable aid for doing the required group theory computations., 38 pages

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We investigate some problems in algebraic coding theory and finite geometry by relating them to polynomials in two variables and applying Weil's theorem. We prove absolute irreducibility of polynomials arising in this way using Bezout's theorem. In Chapter 2 we investigate certain cyclic codes, and we show that there are codewords of a certain weight by proving that some polynomials are absolutely irreducible and applying Weil's theorem. In Chapter 3 we investigate the existence of hyperovals which have the form [...] in finite projective planes of even order, and we show that there must be three collinear points by proving that some polynomials are absolutely irreducible and applying Weil's theorem. In Chapter 4 we discuss Galois rings of order [...]. We construct a relative difference set from these, and hence an affine plane, which we prove is Desarguesian. We also construct binary codes from the Galois rings, and we prove that there are codewords of a certain weight in the natural generalization of the Preparata and Goethals codes by proving that some polynomials are absolutely irreducible and applying Weil's theorem.

Codewords of weight ≤ 4 in certain cyclic codes of length n = 2s − 1, parameterized by an odd integer t, can be related to zeros of certain projective plane curves gt(X, Y, Z). Some families of these codes have been shown to have no codewords of weight ≤ 4, i.e., they are 2-error-correcting codes. But if the polynomials gt(X, Y, Z) are absolutely irreducible, Weil′s theorem shows that the codes do have codewords of weight 4 for all integers s that are sufficiently large with respect to t. Here we prove that gt(X, Y, Z) is, absolutely irreducible for all integers t > 3 such that t ≡ 3(mod 4), and also for some values t ≡ 1(mod 4). These cases provide us with evidence for a conjecture that would classify all such codes in terms of their minimum distance. The methods of proving absolute irreducibility involve Bezout′s theorem and may be of independent interest.

The Dickson–Guralnick–Zieve curve, briefly DGZ curve, defined over the finite field F q arises naturally from the classical Dickson invariant of the projective linear group P G L ( 3 , F q ) . The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q 3 − q 2 and genus 1 2 q ( q − 1 ) ( q 3 − 2 q − 2 ) + 1 . In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse–Witt invariant is positive; the Fermat curve of degree q − 1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over F q 3 , the DGZ curve is optimal with respect the number of its F q 3 -rational points.

Planar functions are mappings from a finite field F q to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q 1 / 4 that induce planar functions on F q , by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over F q that induce planar functions on infinitely many extensions of F q . The proof strategy is to study the number of F q -rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions.

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of \'etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. {\bf 3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on $\mathbb{P}^1$ with logarithmic structure on marked points $D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $\pi_1^\text{et}(\mathbb{P}^1_{{\mathbb{Q}}_p^\text{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,\lambda\}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve $\mathcal{C}_\lambda$ defined by $ y^2=x(x-1)(x-\lambda)$ with the order coprime to $p$., Comment: 84 pages