Rationality of Weil Sums of Binomials with Four Distinct Values

Alapfogalmak

Weil sums of binomials with four distinct values must be rational integers, except in the case where the finite field has order 5 and the exponent is congruent to 3 modulo 4.

Kivonat

The content discusses the rationality of Weil sums of binomials, which are sums of the form W K,s
u = Σ_x∈K ψ(xs - ux), where K is a finite field, ψ is the canonical additive character of K, u is an element of K×, and s is a positive integer relatively prime to |K×|.

The key insights are:

1. The Weil spectrum, which is the set of distinct values that the Weil sums W K,s
   u can take as u runs through K×, always contains at least three distinct values if s is nondegenerate (i.e., not a power of the characteristic p modulo |K×|).

2. It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers.

3. The main result of the paper shows that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where |K| = 5 and s ≡ 3 (mod 4).

The paper uses tools from algebraic number theory, p-adic analysis, and the study of certain algebraic sets over finite fields to prove this result.

Statisztikák

Weil sums W K,s
u are sums of pth roots of unity, where p is the characteristic of the finite field K.
The Weil spectrum for K and s is the multiset of distinct values that the Weil sums W K,s
u can take as u runs through K×.
The Weil spectrum always contains at least three distinct values if s is nondegenerate.
If the Weil spectrum contains precisely three distinct values, then they must all be rational integers.

Idézetek

"If the Weil spectrum for K and s is 4-valued, then it is rational unless K = F5 and s ≡ 3 (mod 4) (in which case WK,s = {(5 ± √5)/2, ± √5})."

Főbb Kivonatok

Rationality of Four-Valued Families of Weil Sums of Binomials

by Daniel J. Ka... : arxiv.org 04-09-2024

https://arxiv.org/pdf/2306.14414.pdf

Mélyebb kérdések

How do the properties of Weil sums relate to applications in arithmetic geometry and information theory

The properties of Weil sums play a crucial role in both arithmetic geometry and information theory. In arithmetic geometry, Weil sums are used to count points on algebraic varieties over finite fields. They provide a way to understand the geometric properties of these varieties and have applications in areas such as elliptic curves, algebraic surfaces, and more. Weil sums help in studying the distribution of points and understanding the structure of these algebraic sets.
In information theory, Weil sums are utilized in various applications, particularly in cryptography and coding theory. They are used to analyze the nonlinearity of functions, which is essential in cryptographic algorithms to ensure security against attacks. Weil sums also have applications in error-correcting codes, where they help determine the weight distribution of codes, indicating their performance in correcting errors in data transmission.
The rationality of Weil sums is significant in both arithmetic geometry and information theory, as it provides insights into the algebraic properties of these sums and their applications in practical scenarios.

What other families of Weil sums, beyond the 4-valued case, can be shown to have rational values

Beyond the 4-valued case, there are several other families of Weil sums that can be shown to have rational values. For example, 3-valued Weil sums have been extensively studied and are known to have rational values. The rationality of Weil sums is often linked to the properties of the underlying finite field and the exponent used in the calculations. In cases where the Weil spectrum contains a small number of distinct values, such as 3 or 4 values, it is more likely that these values will be rational integers.
Additionally, there are specific conditions and criteria, such as the congruence of the exponent with the characteristic of the field, that determine the rationality of Weil sums. By analyzing the structure of the Weil spectrum and the properties of the finite field, researchers can establish the rationality of Weil sums in various scenarios beyond just the 4-valued families.

Are there connections between the rationality of Weil sums and the structure of the underlying finite field and exponent

The rationality of Weil sums is closely connected to the structure of the underlying finite field and the choice of the exponent. The properties of the finite field, such as its characteristic and order, influence the rationality of Weil sums. For example, the congruence of the exponent with the characteristic of the field can determine whether the Weil sums will have rational values.
Moreover, the structure of the finite field impacts the distribution of values in the Weil spectrum. Certain families of Weil sums, particularly those with a small number of distinct values, are more likely to have rational values based on the characteristics of the field. Understanding the algebraic properties of the finite field and the relationships between the exponent and the field's characteristics is essential in determining the rationality of Weil sums.