A Multiple Exponential Sum to Modulus p2

1985 ◽  
Vol 28 (4) ◽  
pp. 394-396 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Total Degree ◽  
Image Position

AbstractFor suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown thatThis is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

2009 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV
Keyword(s):  
Prime Order ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Complex Numbers ◽  
Number Of Solutions ◽  
Prime Fields ◽  
Explicit Bounds ◽  
Multiplicative Subgroup ◽  
Image Position

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
LUCK DARNIÈRE ◽  
EVA LEENKNEGT
Keyword(s):  
Algebraic Geometry ◽  
Open Subset ◽  
Cell Decomposition ◽  
Real Algebraic Geometry ◽  
Dimension Theory ◽  
Definable Set ◽  
Image Position ◽  
Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

On a New Exponential Sum

2001 ◽  
Vol 44 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Daniel Lieman ◽  
Igor Shparlinski
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Multiplicative Order

AbstractLet p be prime and let be of multiplicative order t modulo p. We consider exponential sums of the formand prove that for any ε > 0

Mr. Gibbins′ Triangle

1939 ◽  
Vol 23 (256) ◽  
pp. 360-363
Author(s):  
F. H. V. Gulasekharam
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Analysis ◽  
Image Position

In the Gazette No. 249, Vol. XXII, Mr. Gibbins has deduced some interesting formulae connected with the triangle ABC, of which the side BC is given by L = lx + my +n= 0 and AB, AC by the equation The equations of the circles in § 3 of Mr. Gibbins’ note are very elegant indeed. But, in principle, it does not seem desirable to have recourse to pure geometry in dealing with problems in algebraic geometry. The algebraic analysis is not difficult. It is not necessary to resolve S into factors. Let us assume that S≡vw, where v ≡ l2x + m2y + n2 and w ≡ l3x + m3y + n3.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

The Estimation of Complete Exponential Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 440-454 ◽  
Author(s):  
J. H. Loxton ◽  
R. C. Vaughan
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Integer Coefficients ◽  
Complex Zeros

AbstractThis paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

scholarly journals Dirichlet and Poincaré series

1985 ◽  
Vol 27 ◽  
pp. 39-56 ◽  
Author(s):  
A. Good
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Entire Function ◽  
Cusp Form ◽  
Modular Forms ◽  
Modular Group ◽  
Positive Proportion ◽  
Fundamental Discriminant ◽  
Image Position

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

PROPRIÉTÉS LOCALES DES CHIFFRES DES NOMBRES PREMIERS

2017 ◽  
Vol 18 (1) ◽  
pp. 189-224
Author(s):  
Bruno Martin ◽  
Christian Mauduit ◽  
Joël Rivat
Keyword(s):  
Asymptotic Formula ◽  
Additive Function ◽  
Prime Numbers ◽  
Exponential Sum ◽  
Image Position

Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum $$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$ where the summation runs over prime numbers $p$ and where $\unicode[STIX]{x1D6FD}\in \mathbb{R}$, $k\in \mathbb{Z}$, and $g:\mathbb{N}\rightarrow \mathbb{Z}$ is a strongly $b$-additive function such that $\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$.

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