Equidistribution of Polynomial Sequences

Definition

The equidistribution of polynomial sequences refers to the concept where a sequence generated by evaluating a polynomial at integer values becomes uniformly distributed in a certain space, typically in the context of number theory and algebraic geometry. This idea is significant in understanding how polynomial maps interact with arithmetic structures and contributes to deep insights in areas like Diophantine equations and ergodic theory.

5 Must Know Facts For Your Next Test

1. Equidistribution is often studied using Weyl's criterion, which provides conditions under which a sequence is uniformly distributed.
2. In the context of number fields, equidistribution can be linked to the distribution of primes and their behavior under polynomial mappings.
3. The equidistribution of polynomial sequences has implications for understanding the behavior of rational points on algebraic varieties.
4. This concept is closely related to topics in ergodic theory, where it connects dynamical systems to number theory through the study of action on spaces.
5. The results of equidistribution often rely on techniques from analytic number theory, including estimates for exponential sums.

Review Questions

• How does Weyl's criterion help determine the equidistribution of polynomial sequences?
  • Weyl's criterion provides a mathematical framework that establishes conditions under which a sequence is uniformly distributed. Specifically, it shows that if the sequence of values produced by evaluating a polynomial at integers satisfies certain properties regarding sums of exponential functions, then the sequence can be considered equidistributed. Understanding this criterion is crucial for analyzing how polynomial sequences behave in various mathematical contexts.
• Discuss the connection between equidistribution of polynomial sequences and Diophantine equations.
  • The equidistribution of polynomial sequences plays a significant role in the study of Diophantine equations, as these equations often involve polynomials seeking integer solutions. When polynomial sequences are shown to be equidistributed, it can reveal patterns or distributions of integer solutions across certain domains. This relationship underscores how understanding polynomial behavior aids in solving complex problems in number theory.
• Evaluate the impact of equidistribution on the distribution of rational points on algebraic varieties.
  • The equidistribution of polynomial sequences significantly affects the distribution of rational points on algebraic varieties by providing insights into how these points are spread throughout the variety. When we establish that certain polynomial sequences are equidistributed, we can infer that rational points exhibit similar uniformity in their distribution. This understanding not only helps in mapping the structure of algebraic varieties but also enhances our comprehension of their geometric properties and potential applications in arithmetic geometry.

Related terms

Uniform Distribution: A property of a sequence that indicates it is spread out evenly over a certain interval or space, such that any subinterval has a proportionate number of points from the sequence.

Polynomial Map: A function defined by a polynomial equation that takes input values and transforms them according to the polynomial's coefficients and terms.

Diophantine Equations: Equations that seek integer solutions and often involve polynomial expressions, highlighting the connection between number theory and polynomial properties.