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# Equations in finite fields with restricted solution sets. II (Algebraic equations)

Acta Mathematica Hungarica
Authors: K. Gyarmati and A. Sárközy

## Abstract

Generalizing earlier results, it is shown that if
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}$$ \end{document}
are “large” subsets of a finite field F q, then the equations a + b = cd, resp. ab + 1 = cd can be solved with
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D}$$ \end{document}
. Other algebraic equations with solutions restricted to “large” subsets of F q are also studied. The proofs are based on character sum estimates proved in Part I of the paper.
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# Equations in finite fields with restricted solution sets. I (Character sums)

Acta Mathematica Hungarica
Authors: K. Gyarmati and A. Sárközy

## Abstract

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z p and for h(x) ∈ Z p[x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with aA, bB, xZ p, and for large subsets A, B, C, D of Z p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with aA, bB, cC, dD. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.

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# On the Diophantine equation

Acta Mathematica Hungarica
Authors: Jiagui Luo and Pingzhi Yuan

References  Cao , Z. F. 1990 On the Diophantine equation Chinese Science Bulletin 35

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# Wilson’s functional equation in algebras

Studia Scientiarum Mathematicarum Hungarica
Author: Żywilla Fechner

Dacić, R. , The cosine functional equation for groups, Math. Vesnik , 6 (21) (1969), 339–342. MR 42 #6453 Dacić R

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# Linear equations and sets of integers

Acta Mathematica Hungarica
Author: Tomasz Schoen

. 28 104 – 109 10.1112/jlms/s1-28.1.104 .  Ruzsa , I. Z. 1993 Solving linear equations in sets of integers. I Acta Arith

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# On non-uniformly parabolic functional differential equations

Studia Scientiarum Mathematicarum Hungarica
Authors: László Simon and Willi Jäger

. A Math. Anal. 8 , 35–51. MR 2002b :35084 Díaz, J. I. and Hetzer, G. , A quasilinear functional reaction-diffusion equation arising in climatology, in: PDEs and applications , Dunod

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# Star-quasilinear equational theories of groupoids

Studia Scientiarum Mathematicarum Hungarica
Authors: Petar Đapć, Jaroslav Ježek, and Petar Marković

Đapić, P., Ježek, J., Marković, P., McKenzie, R. and Stanovský, D. , Star-linear equational theories of groupoids, Algebra Universalis , 56 (2007), no. 3–4, 357–397. MR 2008d

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# Asymptotic behavior of the extended Jensen equation

Studia Scientiarum Mathematicarum Hungarica

’s functional equations and approximate algebra homomorphisms, Bull. Korean Math. Soc. , 39 (2002), no. 3, 401–410. MR 2004g :39047 Park W.-G. Generalized Jensen’s functional

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# Polynomial-exponential equations involving multi-recurrences

Studia Scientiarum Mathematicarum Hungarica
Author: Clemens Fuchs

Artin, M. , On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 38 #344 Artin M. On the

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# On a class of equations stemming from various quadrature rules

Acta Mathematica Hungarica
Authors: B. Koclȩga-Kulpa and T. Szostok

.1017/S0308210509001188 .  Ger , J. 2002 On Sahoo-Riedel equations on a real interval Aequationes Math. 63

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