Quadratic forms

Definition

Quadratic forms are homogeneous polynomial functions of degree two in multiple variables. They can be represented in matrix form as $$Q(x) = x^T A x$$, where $x$ is a vector of variables and $A$ is a symmetric matrix. These forms are essential in various areas of mathematics, including number theory and algebraic geometry, particularly in the study of rational points on varieties.

5 Must Know Facts For Your Next Test

1. Quadratic forms can be classified into different types based on their properties, such as positive definite, negative definite, or indefinite.
2. The study of quadratic forms is crucial in understanding the solutions to equations and their representation over various fields, including rationals and p-adics.
3. In the context of the Hasse principle, a quadratic form may represent a rational point if it does so over both real and p-adic numbers.
4. The rank of a quadratic form, determined by the number of linearly independent variables it involves, can influence its ability to represent numbers.
5. The study of discriminants associated with quadratic forms helps in identifying whether the form can represent certain integers or rational points.

Review Questions

• How do quadratic forms relate to the representation of numbers in different number systems?
  • Quadratic forms play a critical role in determining whether specific numbers can be represented by using integer solutions. By analyzing these forms in various number systems, including real numbers and p-adic numbers, mathematicians can employ the Hasse principle to conclude that if a quadratic form represents a number in these systems, it likely represents it over the rationals as well. This shows how quadratic forms serve as a bridge between algebra and number theory.
• Discuss the significance of positive definiteness in quadratic forms and its implications on representation theory.
  • Positive definiteness in quadratic forms implies that the associated matrix has all positive eigenvalues, ensuring that the form takes only positive values for all non-zero input vectors. This property is crucial for understanding which numbers can be represented by the form; for instance, if a quadratic form is positive definite, it can represent zero and all positive integers but not negative integers. Thus, recognizing when a form is positive definite helps mathematicians identify its range of representable values.
• Evaluate how understanding quadratic forms contributes to solving problems related to the Hasse principle in arithmetic geometry.
  • Understanding quadratic forms enhances our ability to tackle problems associated with the Hasse principle by providing insight into how these forms behave across different fields. By exploring their properties, such as definiteness and discriminants, mathematicians can determine whether rational solutions exist. This connection between quadratic forms and the Hasse principle is vital for establishing criteria under which rational points on varieties can be guaranteed based on their representations in local completions.

Related terms

Positive Definite: A property of a quadratic form where it takes only positive values for all non-zero inputs, indicating that the corresponding matrix is symmetric and has all positive eigenvalues.

Lattice Points: Points in a geometric space whose coordinates are all integers, often studied in relation to quadratic forms to determine whether the form can represent certain integer values.

Hasse Principle: A principle stating that a quadratic form represents a rational number if it represents that number in every completion of the rational numbers, linking number theory with geometry.