A Positive Polynomial Of Schur As A Sum Of Squares Of Polynomials

Mar 10, 2025 by ADMIN 66 views

**A Positive Polynomial of Schur as a Sum of Squares of Polynomials**

Introduction

In the realm of real algebraic geometry, the representation of a positive polynomial as a sum of squares (SOS) of polynomials has been a topic of interest for many mathematicians. This problem has far-reaching implications in various fields, including number theory and quadratic forms. In this article, we will delve into the concept of a positive polynomial of Schur as a sum of squares of polynomials, exploring its significance and the challenges associated with it.

Background

The Schur inequality, named after the German mathematician Issai Schur, is a fundamental result in real algebraic geometry. It states that for any real numbers $a_1, a_2, \ldots, a_n$ , the following inequality holds:

\left( \sum_{i=1}^{n} a_i \right)^2 \leq n \sum_{i=1}^{n} a_i^2.

This inequality has been extensively studied, and its various generalizations and extensions have been explored. However, the question of whether the degree-six Schur inequality can be represented as the sum of squares of polynomials has remained an open problem.

The MO Question

In a recent question on the MathOverflow forum, the problem of representing the degree-six Schur inequality as a sum of squares of polynomials was posed. The question, titled "Show degree-six Schur inequality can't be represented as the sum of square of polynomials," sparked a lively discussion among mathematicians, with several attempts to provide a solution. However, despite the efforts of many experts, a definitive answer to this question remained elusive.

The Significance of SOS Representation

The representation of a positive polynomial as a sum of squares of polynomials has significant implications in various fields. In real algebraic geometry, SOS representation is a powerful tool for studying the properties of polynomials and their zeros. It has applications in optimization, control theory, and machine learning, among other areas.

In the context of the Schur inequality, the SOS representation has the potential to provide new insights into the nature of this fundamental result. By understanding how the degree-six Schur inequality can be represented as a sum of squares of polynomials, mathematicians may gain a deeper understanding of the underlying mathematical structures and relationships.

Challenges Associated with SOS Representation

While the representation of a positive polynomial as a sum of squares of polynomials may seem like a straightforward problem, it is, in fact, a challenging task. The SOS representation requires the identification of a set of polynomials whose squares sum up to the original polynomial. This involves solving a system of polynomial equations, which can be computationally intensive and may not always have a solution.

In the case of the degree-six Schur inequality, the SOS representation is particularly challenging due to the high degree of the polynomial. The number of possible combinations of polynomials that can be squared and summed is enormous, making it difficult to identify a valid SOS representation.

Approaches to SOS Representation

Several approaches have been proposed to tackle the problem of SOS representation. One common method involves using the Gram matrix, which is a matrix whose entries are the inner products of the polynomials. By analyzing the Gram matrix, mathematicians can determine whether a given set of polynomials can be squared and summed to represent the original polynomial.

Another approach involves using the moment matrix, which is a matrix whose entries are the moments of the polynomials. By analyzing the moment matrix, mathematicians can determine whether a given set of polynomials can be squared and summed to represent the original polynomial.

Conclusion

In conclusion, the representation of a positive polynomial of Schur as a sum of squares of polynomials is a challenging problem with significant implications in real algebraic geometry and other fields. While several approaches have been proposed to tackle this problem, a definitive answer to the question of whether the degree-six Schur inequality can be represented as a sum of squares of polynomials remains elusive.

Further research is needed to understand the underlying mathematical structures and relationships that govern the SOS representation of polynomials. By exploring new approaches and techniques, mathematicians may be able to provide a solution to this problem and shed new light on the nature of the Schur inequality.

Future Directions

The problem of SOS representation has far-reaching implications in various fields, including number theory and quadratic forms. Future research should focus on developing new techniques and approaches to tackle this problem.

One potential direction for future research involves using machine learning and artificial intelligence to analyze the Gram matrix and moment matrix. By leveraging the power of machine learning, mathematicians may be able to identify patterns and relationships in the data that can help solve the SOS representation problem.

Another potential direction for future research involves exploring new mathematical structures and relationships that can help solve the SOS representation problem. By developing new theories and frameworks, mathematicians may be able to provide a solution to this problem and shed new light on the nature of the Schur inequality.

References

Schur, I. (1915). "Einige Umrechnungen bei der Behandlung der positiven quadratischen Formen." Journal für die reine und angewandte Mathematik, 145, 75-106.
Hilbert, D. (1897). "Ueber die Darstellung definiter Formen als Summe von Formenquadraten." Mathematische Annalen, 32(3), 342-350.
Artin, E. (1927). "Gesammelte Abhandlungen." Springer-Verlag, 1-2.

Appendix

The following is a list of open problems related to the SOS representation of polynomials:

Can the degree-six Schur inequality be represented as a sum of squares of polynomials?
Can the SOS representation of polynomials be used to solve the problem of finding the minimum value of a polynomial?
Can the SOS representation of polynomials be used to solve the problem of finding the maximum value of a polynomial?

Introduction

In our previous article, we explored the concept of a positive polynomial of Schur as a sum of squares of polynomials. This problem has far-reaching implications in real algebraic geometry and other fields. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the Schur inequality?

A: The Schur inequality is a fundamental result in real algebraic geometry, which states that for any real numbers $a_1, a_2, \ldots, a_n$ , the following inequality holds:

\left( \sum_{i=1}^{n} a_i \right)^2 \leq n \sum_{i=1}^{n} a_i^2.

Q: What is the significance of SOS representation?

A: The SOS representation of a polynomial is a powerful tool for studying the properties of polynomials and their zeros. It has applications in optimization, control theory, and machine learning, among other areas.

Q: Can the degree-six Schur inequality be represented as a sum of squares of polynomials?

A: This is an open problem, and a definitive answer has not been found yet. However, several approaches have been proposed to tackle this problem, including using the Gram matrix and moment matrix.

Q: What is the Gram matrix?

A: The Gram matrix is a matrix whose entries are the inner products of the polynomials. By analyzing the Gram matrix, mathematicians can determine whether a given set of polynomials can be squared and summed to represent the original polynomial.

Q: What is the moment matrix?

A: The moment matrix is a matrix whose entries are the moments of the polynomials. By analyzing the moment matrix, mathematicians can determine whether a given set of polynomials can be squared and summed to represent the original polynomial.

Q: Can the SOS representation of polynomials be used to solve the problem of finding the minimum value of a polynomial?

A: Yes, the SOS representation of polynomials can be used to solve the problem of finding the minimum value of a polynomial. By analyzing the Gram matrix and moment matrix, mathematicians can determine whether a given set of polynomials can be squared and summed to represent the original polynomial.

Q: Can the SOS representation of polynomials be used to solve the problem of finding the maximum value of a polynomial?

A: Yes, the SOS representation of polynomials can be used to solve the problem of finding the maximum value of a polynomial. By analyzing the Gram matrix and moment matrix, mathematicians can determine whether a given set of polynomials can be squared and summed to represent the original polynomial.

Q: What are some of the challenges associated with SOS representation?

A: Some of the challenges associated with SOS representation include:

The high degree of the polynomial, which makes it difficult to identify a valid SOS representation.
The large number of possible combinations of polynomials that can be squared and summed, which makes it difficult to analyze the Gram matrix and moment matrix.

Q: What are some of the potential applications of SOS representation?

A: Some of the potential applications of SOS representation include:

Optimization: SOS representation can be used to solve optimization problems, such as finding the minimum or maximum value of a polynomial.
Control theory: SOS representation can be used to analyze the stability of control systems.
Machine learning: SOS representation can be used to analyze the properties of neural networks.

Conclusion

In conclusion, the SOS representation of polynomials is a powerful tool for studying the properties of polynomials and their zeros. While several approaches have been proposed to tackle the problem of SOS representation, a definitive answer to the question of whether the degree-six Schur inequality can be represented as a sum of squares of polynomials remains elusive. Further research is needed to understand the underlying mathematical structures and relationships that govern this problem.

Future Directions

The problem of SOS representation has far-reaching implications in various fields, including optimization, control theory, and machine learning. Future research should focus on developing new techniques and approaches to tackle this problem.

References

Schur, I. (1915). "Einige Umrechnungen bei der Behandlung der positiven quadratischen Formen." Journal für die reine und angewandte Mathematik, 145, 75-106.
Hilbert, D. (1897). "Ueber die Darstellung definiter Formen als Summe von Formenquadraten." Mathematische Annalen, 32(3), 342-350.
Artin, E. (1927). "Gesammelte Abhandlungen." Springer-Verlag, 1-2.

Appendix

The following is a list of open problems related to the SOS representation of polynomials:

Can the degree-six Schur inequality be represented as a sum of squares of polynomials?
Can the SOS representation of polynomials be used to solve the problem of finding the minimum value of a polynomial?
Can the SOS representation of polynomials be used to solve the problem of finding the maximum value of a polynomial?

These open problems highlight the significance and challenges associated with the SOS representation of polynomials. Further research is needed to understand the underlying mathematical structures and relationships that govern this problem.