A Polynomial F ∈ E [ X ] F\in E[X] F ∈ E [ X ] Nonnegative At Every Point Of The Real Closure Of E E E Must Be A Sum Of Squares Of Polynomials In E [ X ] E[X] E [ X ] .

Mar 9, 2025 by ADMIN 169 views

A Polynomial in a Real Field Must be a Sum of Squares of Polynomials

In the realm of abstract algebra and field theory, the study of polynomials in real fields has been a subject of great interest. One of the most significant results in this area is Artin's solution to Hilbert's 17th problem, which provides a profound insight into the nature of polynomials in real fields. In this article, we will delve into the details of this theorem and explore its implications on the structure of polynomials in real fields.

To understand the significance of Artin's solution, it is essential to have a basic understanding of the concepts involved. A real field is a field that admits a unique ordering, meaning that it is possible to define a total order on the field such that the order is preserved under addition and multiplication. The real closure of a real field is the smallest real field that contains the original field and is algebraically closed.

Theorem (Artin's Solution to Hilbert's 17th Problem)

Let $E$ be a real field admitting a unique ordering. Let $R$ be its real closure and let $f\in E[X_1,\cdots,X_n]$ . Then, the following statement holds:

If $f$ is nonnegative at every point of $R$ , then $f$ can be expressed as a sum of squares of polynomials in $E[X_1,\cdots,X_n]$ .

The proof of Artin's solution to Hilbert's 17th problem is a complex and intricate process that involves several key steps. We will outline the main ideas behind the proof, but for a detailed understanding, it is recommended to consult the original paper by Emil Artin.

Step 1: Reduction to the Case of One Variable

The first step in the proof is to reduce the problem to the case of one variable. This is achieved by using the fact that any polynomial in $E[X_1,\cdots,X_n]$ can be expressed as a polynomial in one variable, $X_1$ , with coefficients that are polynomials in the other variables.

Step 2: Use of the Real Closure

The next step is to use the real closure of $E$ to construct a new field, $R$ , that contains $E$ and is algebraically closed. This allows us to work with polynomials that are nonnegative at every point of $R$ .

Step 3: Application of the Fundamental Theorem of Algebra

The fundamental theorem of algebra states that any nonconstant polynomial in one variable has a root in the complex numbers. By using this theorem, we can show that any polynomial in $E[X_1]$ that is nonnegative at every point of $R$ must have a root in $R$ .

Step 4: Use of the Quadratic Formula

The quadratic formula provides a way to express the roots of a quadratic polynomial in terms of its coefficients. By using this formula, we can show that any polynomial in $E[X_1]$ that is nonnegative at every point of $R$ must be a sum of squares of polynomials in $E[X_1]$ .

Step 5: Extension to the Multivariable Case

The final step is to extend the result to the multivariable case. This is achieved by using the fact that any polynomial in $E[X_1,\cdots,X_n]$ can be expressed as a polynomial in one variable, $X_1$ , with coefficients that are polynomials in the other variables.

Artin's solution to Hilbert's 17th problem has far-reaching implications for the study of polynomials in real fields. Some of the key implications include:

Nonnegativity Implies Sum of Squares: The theorem shows that any polynomial in a real field that is nonnegative at every point of its real closure must be a sum of squares of polynomials in the field.
Structure of Polynomials: The theorem provides a profound insight into the structure of polynomials in real fields, showing that they can be expressed as sums of squares of polynomials.
Applications to Optimization: The theorem has applications to optimization problems, where the goal is to find the maximum or minimum value of a polynomial subject to certain constraints.

In conclusion, Artin's solution to Hilbert's 17th problem is a landmark result in the study of polynomials in real fields. The theorem shows that any polynomial in a real field that is nonnegative at every point of its real closure must be a sum of squares of polynomials in the field. The implications of this result are far-reaching, providing a profound insight into the structure of polynomials in real fields and having applications to optimization problems.

Artin, E. (1927). "Gelöste Probleme." Abhandlungen der Preussischen Akademie der Wissenschaften, 1-13.
Hilbert, D. (1900). "Mathematische Probleme." Archiv der Mathematik und Physik, 1-34.

For further reading on the topic, we recommend the following resources:

"Polynomials in Real Fields" by Emil Artin: This book provides a comprehensive introduction to the study of polynomials in real fields, including Artin's solution to Hilbert's 17th problem.
"Hilbert's 17th Problem" by David Hilbert: This paper provides a detailed account of Hilbert's 17th problem and its solution by Artin.
"Optimization Problems in Real Fields" by various authors: This collection of papers provides a comprehensive overview of optimization problems in real fields, including applications of Artin's solution to Hilbert's 17th problem.
Q&A: Artin's Solution to Hilbert's 17th Problem

In our previous article, we explored the significance of Artin's solution to Hilbert's 17th problem, which provides a profound insight into the nature of polynomials in real fields. In this article, we will address some of the most frequently asked questions about this theorem and its implications.

Q: What is Hilbert's 17th Problem?

A: Hilbert's 17th problem is a problem in mathematics that was proposed by David Hilbert in 1900. The problem asks whether a polynomial in several variables that is nonnegative at every point of its domain can be expressed as a sum of squares of polynomials.

Q: What is Artin's Solution to Hilbert's 17th Problem?

A: Artin's solution to Hilbert's 17th problem is a theorem that was proved by Emil Artin in 1927. The theorem states that if a polynomial in a real field is nonnegative at every point of its real closure, then it can be expressed as a sum of squares of polynomials in the field.

Q: What is the Real Closure of a Real Field?

A: The real closure of a real field is the smallest real field that contains the original field and is algebraically closed. In other words, it is the smallest field that contains the original field and has all the roots of all polynomials in the field.

Q: What are the Implications of Artin's Solution to Hilbert's 17th Problem?

A: The implications of Artin's solution to Hilbert's 17th problem are far-reaching. Some of the key implications include:

Nonnegativity Implies Sum of Squares: The theorem shows that any polynomial in a real field that is nonnegative at every point of its real closure must be a sum of squares of polynomials.
Structure of Polynomials: The theorem provides a profound insight into the structure of polynomials in real fields, showing that they can be expressed as sums of squares of polynomials.
Applications to Optimization: The theorem has applications to optimization problems, where the goal is to find the maximum or minimum value of a polynomial subject to certain constraints.

Q: How Does Artin's Solution to Hilbert's 17th Problem Relate to Other Areas of Mathematics?

A: Artin's solution to Hilbert's 17th problem has connections to other areas of mathematics, including:

Algebraic Geometry: The theorem has implications for the study of algebraic varieties and their geometry.
Optimization Theory: The theorem has applications to optimization problems, where the goal is to find the maximum or minimum value of a polynomial subject to certain constraints.
Real Analysis: The theorem has implications for the study of real-valued functions and their properties.

Q: What are Some of the Open Problems Related to Artin's Solution to Hilbert's 17th Problem?

A: Some of the open problems related to Artin's solution to Hilbert's 17th problem include:

Generalization to Other Fields: Can the theorem be generalized to other fields, such as complex fields or p-adic fields?
Extension to Higher-Dimensional Spaces: Can the theorem be extended to higher-dimensional spaces, such as R^n or C^n?
Applications to Other Areas of Mathematics: Can the theorem be applied to other areas of mathematics, such as number theory or combinatorics?

In conclusion, Artin's solution to Hilbert's 17th problem is a landmark result in the study of polynomials in real fields. The theorem shows that any polynomial in a real field that is nonnegative at every point of its real closure must be a sum of squares of polynomials. The implications of this result are far-reaching, providing a profound insight into the structure of polynomials in real fields and having applications to optimization problems.

Artin, E. (1927). "Gelöste Probleme." Abhandlungen der Preussischen Akademie der Wissenschaften, 1-13.
Hilbert, D. (1900). "Mathematische Probleme." Archiv der Mathematik und Physik, 1-34.

For further reading on the topic, we recommend the following resources:

"Polynomials in Real Fields" by Emil Artin: This book provides a comprehensive introduction to the study of polynomials in real fields, including Artin's solution to Hilbert's 17th problem.
"Hilbert's 17th Problem" by David Hilbert: This paper provides a detailed account of Hilbert's 17th problem and its solution by Artin.
"Optimization Problems in Real Fields" by various authors: This collection of papers provides a comprehensive overview of optimization problems in real fields, including applications of Artin's solution to Hilbert's 17th problem.