Equations over finite fields. An elementary approach

DOI10.1007/BFb0080437zbMath0329.12001OpenAlexW1489774662MaRDI QIDQ1227023

Wolfgang M. Schmidt

Publication date: 1976

Published in: Lecture Notes in Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bfb0080437

zbMATH Keywords

exponential sumscharacter sumsabsolutely irreducible equationsequations in many variablesnumber of points in varieties over finite fields

Mathematics Subject Classification ID

Research exposition (monographs, survey articles) pertaining to number theory (11-02) Curves over finite and local fields (11G20) Polynomials over finite fields (11T06) Exponential sums (11T23) Finite ground fields in algebraic geometry (14G15) Other character sums and Gauss sums (11T24) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) (14G10) Higher degree equations; Fermat's equation (11D41) Varieties over finite and local fields (11G25) Arithmetic theory of polynomial rings over finite fields (11T55)

Related Items

Using Stepanov's method for exponential sums involving rational functions, Fast parallel absolute irreducibility testing, The correlation between parity and quadratic polynomials mod \(3\), Class numbers of quadratic extensions of algebraic number fields, The Markoff group of transformations in prime and composite moduli. With an appendix by Dan Carmon., Derandomizing restricted isometries via the Legendre symbol, Large supremum norms and small Shannon entropy for Hecke eigenfunctions of quantized cat maps, On subgroups of \(GL_ n(F_ p)\), Families of polynomials and their specializations, Long binary narrow-sense BCH codes are normal, Further improvements in Waring's problem, On the reduction modulo \(p\) of an absolutely irreducible polynomial \(f(x,y)\), Monochromatic sums of squares, Cubic and quadruple Paley graphs with the \(n\)-e.c. property, Triangles and elliptic curves. IV, Lower bound of overall exponential sums for polynomials \(\phi(x^ d)\), On 6-sparse Steiner triple systems, Counting points in a small box on varieties, On small zeros of quadratic forms over finite fields, Friability of the values of a polynomial, Extractors for varieties, Dickson curves, Nonsingular points on hypersurfaces over \(\mathbb F_q\), The fourth and sixth power mean of the classical Kloosterman sums, On low weight codewords of generalized affine and projective Reed-Muller codes, Divisibility of the class numbers of imaginary quadratic fields, Prime and almost prime integral points on principal homogeneous spaces, Upper bounds on character sums with rational function entries, Irreducibility of polynomials modulo \(p\) via Newton polytopes., The number of reducible space curves over a finite field, Hecke eigenfunctions of quantized cat maps modulo prime powers, On additive decompositions of the set of primitive roots modulo \(p\), On the pseudo-randomness of subsets related to primitive roots, Linear complexity profile of \(m\)-ary pseudorandom sequences with small correlation measure, The minimum distance of parameterized codes on projective tori, On the deterministic complexity of factoring polynomials over finite fields, On cubic polynomials. III: Systems of p-adic equations, Factoring polynomials using fewer random bits, On the classification of semifield flocks., On the constant in Burgess' bound for the number of consecutive residues or non-residues, A note on Burgess's estimate, Estimates of character sums in finite fields, Classi di isomorfismo delle cubiche di \(F_q\)., On the sixth power mean value of the generalized three-term exponential sums, A hybrid mean value of Dedekind sums and Kloosterman sums, Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field, Affine distance-transitive groups of dimension one, A note on certain modular constructions of pseudorandom binary sequences with composite moduli, The distribution of spacings between quadratic residues, On the complexity of families of pseudo-random subsets, On a conjecture of Fernando, Hou and Lappano concerning permutation polynomials over finite fields, The fourth power mean of the general 3-dimensional Kloostermann sums mod \(p\), A class of self-orthogonal 2-sequencings, Reducibility of rational functions in several variables, Equations in finite fields with restricted solution sets. I: Character sums, Trace of products in finite fields, Prescribing digits in finite fields, Unimodular graphs and Eisenstein sums, Artin's primitive root conjecture for function fields revisited, The existence of 5-sparse Steiner triple systems of order \(n \equiv 3 \mod 6\), \(n \notin \{9,15 \}\), Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes, Covering arrays from cyclotomy, An application of exponential sum estimates, On large families of subsets of the set of the integers not exceeding \(N\), A reciprocity formula for quadratic forms, Existence of primitive 2-normal elements in finite fields, Average distribution of prime ideals in families of number fields, An average bound for character sums with some counter-dependent recurrence sequences, On the sum of digits of special sequences in finite fields, On explicit random-like tournaments, Applications of the Hasse-Weil bound to permutation polynomials, Incomplete sums, DC-constrained codes, and codes that maintain synchronization, On the spectrum of projective norm-graphs, Simultaneous solutions to diagonal equations over finite fields, On the exponents of class groups of some families of imaginary quadratic fields, On pseudorandom sequences of \(k\) symbols constructed using finite fields, On the pseudorandom properties of subsets constructed by using primitive roots, Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields, Interpolation of the discrete logarithm in \(\mathbb{F}_{q}\) by Boolean functions and by polynomials in several variables modulo a divisor of \(q-1\)., Arcs and ovals in the Hermitian and Ree unitals, On two exponential sums and their applications, On pseudo-random subsets of \({\mathbb{Z}}_n\), Linearly recurring solution sequences for equations over finite fields, Factoring polynomials over arbitrary finite fields, Zeros of a pair of quadratic forms defined over a finite field, Codes on fibre products of some Kummer coverings, Further 6-sparse Steiner triple systems, Finite Euclidean graphs and Ramanujan graphs, Construction of pseudorandom binary sequences using additive characters over \(GF(2^k)\), Some graphs with small second eigenvalue, Reducibility mod \(p\) of hypersurfaces in projective spaces -- an application of arithmetic Bézout, On some classes of Hilbertian fields, Character sum constructions of constrained error-correcting codes, Likeable functions in finite fields, The decoding of extended Reed-Solomon codes, Degree 8 maximal arcs in PG\((2,2^{h}\)), \(h\) odd, On Weil's proof of the bound for Kloosterman sums., Incomplete character sums and polynomial interpolation of the discrete logarithm, On character sums and codes, Probabilistic absolute irreducibility test for polynomials, ON WARING’S PROBLEM IN SUMS OF THREE CUBES FOR SMALLER POWERS, Nearly sharp Lang–Weil bounds for a hypersurface, ON WARING'S PROBLEM IN SUMS OF THREE CUBES, Uniform bounds for rational points on hyperelliptic fibrations, Consecutive quadratic residues in Beatty sequences, Additive averages of multiplicative correlation sequences and applications, On the eigenvalues of the graphs \(D(5,q)\), Divisibility properties of polynomial expressions of random integers, On the distribution of primitive roots and Lehmer numbers, On Hasse's inequality, Fourth power mean values of one kind special Kloosterman's sum, Unnamed Item, Construction of large families of pseudorandom binary sequences, Factoring polynomials over finite fields: A survey, Consequences of a result of N. Katz and G. Laumon concerning trigonometric sums, Constructive lower bounds for off-diagonal Ramsey numbers, Absolute irreducibility of polynomials via Newton polytopes, A trace conjecture and flag-transitive affine planes, Towards van der Waerden’s conjecture, On the number of zeros of diagonal quartic forms over finite fields, Character sums and small eigenvalues for Г₀(p), One kind sixth power mean of the three-term exponential sums, A note on the moments of Kloosterman sums, The recurrence formula for the number of solutions of a equation in finite field, The number of rational points of certain quartic diagonal hypersurfaces over finite fields, On Kemnitz' conjecture concerning lattice-points in the plane, On the adjacency properties of paley graphs, On \(\mathcal V\)-Weierstrass sets and gaps, Matrix Waring problem, Integration of Algebraic Functions, On the subspace choosability in graphs, Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups, Normality of the Thue-Morse function for finite fields along polynomial values, Random algebraic construction of extremal graphs, On expressing elements as a sum of squares, where one square is restricted to a subfield, Normal bases and primitive elements over finite fields, Integral points on Markoff type cubic surfaces, Consecutive quadratic residues and primitive roots in the sequences formed by twice-differentiable functions, Maximal operators associated to multiplicative characters, Deterministic extractors for affine sources over large fields, REPRESENTING AN ELEMENT IN AS THE SUM OF TWO IRREDUCIBLES, A Partition Result for Algebraic Varieties, A note on graphs with a prescribed adjacency property, The least quadratic non-residue, Simple Constructions of Almost k-wise Independent Random Variables, Necessary and sufficient conditions of two classes of permutation polynomials, Partial Gaussian sums III, Simultaneous additive congruences to a large prime modulus, On the exceptional set for the sum of a prime and ak‐th power, The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and \(U_q(\mathrm{sl}_2)\), Variations of the primitive normal basis theorem, Estimates for character sums in finite fields of order \(p^2\) and \(p^3\), Generic solutions of equations with iterated exponentials, On pseudorandom properties of some Dirichlet characters, Permutation polynomials over finite fields, BEYOND ARTIN'S CONJECTURE FOR CUBIC FORMS, A subconvexity bound for Hecke L-functions, On the symmetry measure of pseudorandom subsets, A note on Weil's multiplicative character sum, On the correlation of symmetric functions, Affine Cartesian codes, The Mathematical Work of Wolfgang Schmidt, The coefficients of primitive polynomials over finite fields, Diophantine equations and congruences over function fields, On the reducibility of large sets of residues modulo \(p\), Quadratic residues and non-residues in arithmetic progression, An explicit construction of \((3,t)\)-existentially closed graphs, Integer points on elliptic curves, Low-lying zeros of families of elliptic curves, The convex hull of the lattice points inside a curve, On generating functions in additive number theory. II: Lower-order terms and applications to PDEs, A Pair of Quadratic Fields with Class Number Divisible by 3, Bounds of multiplicative character sums over shifted primes, An improved estimate of the fourth power mean of the general \(3\)-dimensional Kloosterman sum mod \(p\), Minimum distance functions of complete intersections, Distribution of rational points on varieties over finite fields, Small solutions of congruences, On the Distribution of Quadratic Residues and Nonresidues Modulo a Prime Number, The least \(k\)-th power non-residue, An extension of the (strong) primitive normal basis theorem, Forms of low degree in finite fields, On simultaneous additive equations, III, Sur certaines sommes d'exponentielles sur les nombres premiers, Rational solutions of pairs of diagonal equations, one cubic and one quadratic, Improved explicit estimates on the number of solutions of equations over a finite field, Some results on counting roots of polynomials and the Sylvester resultant, Realistic analysis of some randomized algorithms, Packings in Real Projective Spaces, A SPECIAL DEGREE REDUCTION OF POLYNOMIALS OVER FINITE FIELDS WITH APPLICATIONS, On mean values of character sums, L'ensemble exceptionnel dans la conjecture de Szpiro, Infinite Paley graphs, Permutation Polynomials and Resolution of Singularities over Finite Fields, Sums of Powers in Large Finite Fields: A Mix of Methods, On the correlation of symmetric functions, Survey of Spectra of Laplacians on Finite Symmetric Spaces, Correlation of arithmetic functions over \(\mathbb{F}_q[T\)], A_m-permutation polynomials, Improved bounds for quantified derandomization of constant-depth circuits and polynomials, Existence results on \(k\)-normal elements over finite fields, A note on the least quadratic non-residue of the integer-sequences, The distribution of rational points on varieties defined over a finite field, The determination of Gauss sums, On a family of pseudorandom binary sequences, Zeros of Diagonal Equations over Finite Fields, Fast computation of a rational point of a variety over a finite field, A generalization of the Stein-Lorenzini inequality., DEGREE MATRICES AND ESTIMATES FOR EXPONENTIAL SUMS OF POLYNOMIALS OVER FINITE FIELDS, Simultaneous diagonal p –adic equations, Specializations of Finitely Generated Subgroups of Abelian Varieties, Specializations of Finitely Generated Subgroups of Abelian Varieties, The Furstenberg-Sárközy theorem and asymptotic total ergodicity phenomena in modular rings, Trinomials, singular moduli and Riffaut's conjecture, Control of the non-geometrically integral reductions, Simultaneous cubic and quadratic diagonal equations in 12 prime variables, Ample simplicial complexes, Infinite families of class groups of quadratic fields with 3-rank at least one: quantitative bounds, Diameters and Eigenvalues