10 results on '"Absolutely irreducible"'
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2. Classification of permutation polynomials of the form x3g(xq-1)Fq2g(x)=x3+bx+cb,c∈Fq∗ of x3g(xq-1)Fq2g(x)=x3+bx+cb,c∈Fq∗ where x3g(xq-1)Fq2g(x)=x3+bx+cb,c∈Fq∗ and x3g(xq-1)Fq2g(x)=x3+bx+cb,c∈Fq∗
- Author
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Özbudak, Ferruh and Gülmez Temür, Burcu
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- 2022
- Full Text
- View/download PDF
3. On the roots of certain Dickson polynomials.
- Author
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Blokhuis, Aart, Cao, Xiwang, Chou, Wun-Seng, and Hou, Xiang-Dong
- Subjects
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DICKSON polynomials , *INTEGERS , *FINITE fields , *CONTINUOUS functions , *MATHEMATICAL analysis - Abstract
Let n be a positive integer, q = 2 n , and let F q be the finite field with q elements. For each positive integer m , let D m ( X ) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1 . We study the existence of α ∈ F q ⁎ such that D m ( α ) = D m ( α − 1 ) = 0 . We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”. [ABSTRACT FROM AUTHOR]
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- 2018
- Full Text
- View/download PDF
4. The k-subset sum problem over finite fields.
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Wang, Weiqiong and Nguyen, Jennifer
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SUBSET selection , *FINITE fields , *ALGEBRAIC fields , *ORDERED algebraic structures , *CODING theory , *CRYPTOGRAPHY , *GRAPH theory - Abstract
The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k -subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k -subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results. [ABSTRACT FROM AUTHOR]
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- 2018
- Full Text
- View/download PDF
5. Algebraic Geometry Codes Over Abelian Surfaces Containing No Absolutely Irreducible Curves Of Low Genus
- Author
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Elena Berardini, Yves Aubry, Fabien Herbaut, Marc Perret, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Université Côte d'Azur (UCA), École supérieure du professorat et de l'éducation - Académie de Nice (ESPE Nice), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-Université Côte d'Azur (UCA), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
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FOS: Computer and information sciences ,Pure mathematics ,Absolutely irreducible ,Computer Science - Information Theory ,Abelian surface ,0102 computer and information sciences ,Algebraic geometry ,01 natural sciences ,Theoretical Computer Science ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics ,Algebra and Number Theory ,Applied Mathematics ,Information Theory (cs.IT) ,010102 general mathematics ,Minimum distance ,General Engineering ,[MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT] ,Elliptic curve ,Finite field ,010201 computation theory & mathematics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] - Abstract
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.
- Published
- 2019
6. On the Dickson–Guralnick–Zieve curve
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Massimo Giulietti, Marco Timpanella, and Gábor Korchmáros
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Algebra and Number Theory ,Absolutely irreducible ,Plane curve ,Algebraic curves ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Finite field ,Finite fields ,Automorphism groups ,Fermat curve ,Projective linear group ,Algebraic curve ,0101 mathematics ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
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.
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- 2019
7. On the roots of certain Dickson polynomials
- Author
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Wun-Seng Chou, Xiwang Cao, Xiang-dong Hou, Aart Blokhuis, Discrete Mathematics, and Discrete Algebra and Geometry
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Algebra and Number Theory ,Degree (graph theory) ,Dickson polynomials ,Absolutely irreducible ,Divisor ,010102 general mathematics ,Dickson polynomial ,Reciprocal polynomial ,Fermat number ,Finite field ,0102 computer and information sciences ,Button madness ,01 natural sciences ,Combinatorics ,Integer ,010201 computation theory & mathematics ,0101 mathematics ,Mathematics - Abstract
Let n be a positive integer, q = 2 n , and let F q be the finite field with q elements. For each positive integer m, let D m ( X ) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1 . We study the existence of α ∈ F q ⁎ such that D m ( α ) = D m ( α − 1 ) = 0 . We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.
- Published
- 2018
8. Projective Crystalline Representations of \'Etale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow
- Author
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Jinbang Yang, Ruiran Sun, and Kang Zuo
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Pure mathematics ,Degree (graph theory) ,Coprime integers ,Mathematics - Number Theory ,Absolutely irreducible ,Applied Mathematics ,General Mathematics ,Image (category theory) ,Order (ring theory) ,Higgs bundle ,Higgs field ,Mathematics - Algebraic Geometry ,p-adic Hodge theory ,Mathematics - Abstract
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
- Published
- 2017
9. On the maximum number of rational points on singular curves over finite fields
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Yves Aubry, Annamaria Iezzi, Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Institut de Mathématiques de Marseille (I2M), and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)
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Pure mathematics ,Absolutely irreducible ,General Mathematics ,Geometric genus ,MSC[2010] : 14H20, 11G20, 14G15 ,01 natural sciences ,Mathematics - Algebraic Geometry ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,Arithmetic genus ,FOS: Mathematics ,030212 general & internal medicine ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Singular curves ,010102 general mathematics ,Riemann zeta function ,zeta function ,rational points ,Finite field ,symbols ,Algebraic curve ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,finite fields - Abstract
International audience; We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.
- Published
- 2015
10. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$
- Author
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Eric Férard, Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information (GAATI), and Université de la Polynésie Française (UPF)
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Polynomial (hyperelastic model) ,Fermat's Last Theorem ,Mathematics::Dynamical Systems ,Algebra and Number Theory ,Computer Networks and Communications ,Absolutely irreducible ,Applied Mathematics ,Mathematical analysis ,Combinatorics ,Finite field ,Integer ,Hyperplane ,Discrete Mathematics and Combinatorics ,Irreducibility ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
Let $t$ be an integer $\ge 5$. The absolute irreducibility of the polynomial $\phi_t(x, y) = \frac{x^t + y^t + 1 + (x + y + 1)^t}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb{F}_2$) plays an important role in the study of APN functions. If $t \equiv 5 \bmod{8}$, we give a criterion that ensures that $\phi_t(x, y)$ is absolutely irreducible. We prove that if $\phi_t(x, y)$ is not absolutely irreducible, then it is divisible by $\phi_{13}(x, y)$. We also exhibit an infinite family of integers $t$ such that $\phi_t(x, y)$ is not absolutely irreducible.
- Published
- 2014
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