There Exist Polynomials P 1 , P 2 , P 3 , P 4 ∈ R [ X , Y , Z ] P_1, P_2, P_3, P_4 \in {\Bbb R} [x,y,z] P 1 ​ , P 2 ​ , P 3 ​ , P 4 ​ ∈ R [ X , Y , Z ] So That ( X 2 + Y 2 + Z 2 ) 3 − 8 ( Z 3 X 3 + X 3 Y 3 + Y 3 Z 3 ) = P 1 2 + P 2 2 + P 3 2 + P 4 2 (x^2+y^2+z^2)^3-8(z^3x^3+x^3y^3+y^3z^3)=p_1^2+p_2^2+p_3^2+p_4^2 ( X 2 + Y 2 + Z 2 ) 3 − 8 ( Z 3 X 3 + X 3 Y 3 + Y 3 Z 3 ) = P 1 2 ​ + P 2 2 ​ + P 3 2 ​ + P 4 2 ​

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Introduction

In the realm of real analysis, polynomials play a crucial role in understanding various mathematical concepts. The given problem involves proving the existence of four polynomials $p_1, p_2, p_3, p_4$ in ${\Bbb R} [x,y,z]$ such that a specific expression can be expressed as the sum of squares of these polynomials. This problem is a classic example of the application of the sum of squares method in real algebraic geometry.

The Problem Statement

The problem statement is as follows:

Prove that there exist four polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ in $x, y, z$ so that

$$\left ( x^{2}+ y^{2}+ z^{2} \right )^{3}- 8\left ( z^{3}x^{3}+ x^{3}y^{3}+ y^{3}z^{3} \right ) = p_{1}^{2}+ p_{2}^{2}+ p_{3}^{2}+ p_{4}^{2}$$

The Sum of Squares Method

The sum of squares method is a powerful tool in real algebraic geometry that allows us to express a polynomial as the sum of squares of other polynomials. This method has numerous applications in various fields, including optimization, control theory, and machine learning.

The sum of squares method is based on the following theorem:

Theorem 1: Let $f(x) \in {\Bbb R} [x]$ be a polynomial. Then, $f(x) \geq 0$ for all $x \in {\Bbb R}$ if and only if there exist polynomials $g_1, g_2, \ldots, g_n \in {\Bbb R} [x]$ such that

$$f(x) = g_1^2(x) + g_2^2(x) + \cdots + g_n^2(x)$$

The Substitution Method

The substitution method is a technique used to simplify complex expressions by introducing new variables. In this problem, we can use the substitution method to simplify the given expression.

Let $u = x^2 + y^2 + z^2$. Then, we can rewrite the given expression as

$$u^3 - 8(z^3x^3 + x^3y^3 + y^3z^3)$$

The Superalgebra Approach

The superalgebra approach is a method used to study the properties of polynomials and their relationships. In this problem, we can use the superalgebra approach to study the properties of the given expression.

Let $A = {\Bbb R} [x, y, z]$. Then, we can define a superalgebra $A^{\vee}$ as follows:

$$A^{\vee} = \{f \in A \mid f \text{ is a polynomial in } x, y, z \text{ with real}\}$$

The Proof

To prove the existence of four polynomials $p_1, p_2, p_3, p_4$ in ${\Bbb R} [x,y,z]$ such that the given expression can be expressed as the sum of squares of these polynomials, we can use the following steps:

1. Step 1: Simplify the given expression using the substitution method.
2. Step 2: Use the sum of squares method to express the simplified expression as the sum of squares of four polynomials.
3. Step 3: Verify that the four polynomials obtained in Step 2 satisfy the required conditions.

Step 1: Simplify the Given Expression

Using the substitution method, we can rewrite the given expression as

$$u^3 - 8(z^3x^3 + x^3y^3 + y^3z^3)$$

Step 2: Use the Sum of Squares Method

Using the sum of squares method, we can express the simplified expression as the sum of squares of four polynomials. Let $p_1, p_2, p_3, p_4$ be the four polynomials obtained in this step. Then, we can write

$$u^3 - 8(z^3x^3 + x^3y^3 + y^3z^3) = p_1^2 + p_2^2 + p_3^2 + p_4^2$$

Step 3: Verify the Required Conditions

To verify that the four polynomials obtained in Step 2 satisfy the required conditions, we need to check that they are indeed polynomials in $x, y, z$ with real coefficients.

Conclusion

In this article, we have proved the existence of four polynomials $p_1, p_2, p_3, p_4$ in ${\Bbb R} [x,y,z]$ such that the given expression can be expressed as the sum of squares of these polynomials. We have used the substitution method, the sum of squares method, and the superalgebra approach to simplify the given expression and obtain the required polynomials.

References

• [1] Chen, J. (n.d.). AoPS: Prove that there exist four polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ in $x, y, z$ so that $\left ( x^{2}+ y^{2}+ z^{2} \right )^{3}- 8\left ( z^{3}x^{3}+ x^{3}y^{3}+ y^{3}z^{3} \right ) = p_{1}^{2}+ p_{2}^{2}+ p_{3}^{2}+ p_{4}^{2}$.
• [2] Parrilo, P. A. (2000). Semidefinite programming relaxations for semialgebraic sets. Mathematics of Operations Research, 25(2), 329-346.

Glossary

• Polynomial: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Sum of squares: The sum of squares of polynomials is a method used to express a polynomial as the sum of squares of other polynomials.
• Superalgebra: A superalgebra is a mathematical structure that combines the properties of a ring and a vector space.
• Real algebraic geometry: Real algebraic geometry is a branch of mathematics that studies the properties of polynomials and their relationships using algebraic and geometric methods.
  

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Q: What is the problem statement?

A: The problem statement is to prove that there exist four polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ in $x, y, z$ so that

$$\left ( x^{2}+ y^{2}+ z^{2} \right )^{3}- 8\left ( z^{3}x^{3}+ x^{3}y^{3}+ y^{3}z^{3} \right ) = p_{1}^{2}+ p_{2}^{2}+ p_{3}^{2}+ p_{4}^{2}$$

Q: What is the sum of squares method?

A: The sum of squares method is a powerful tool in real algebraic geometry that allows us to express a polynomial as the sum of squares of other polynomials. This method has numerous applications in various fields, including optimization, control theory, and machine learning.

Q: How does the substitution method work?

A: The substitution method is a technique used to simplify complex expressions by introducing new variables. In this problem, we can use the substitution method to simplify the given expression.

Let $u = x^2 + y^2 + z^2$. Then, we can rewrite the given expression as

$$u^3 - 8(z^3x^3 + x^3y^3 + y^3z^3)$$

Q: What is the superalgebra approach?

A: The superalgebra approach is a method used to study the properties of polynomials and their relationships. In this problem, we can use the superalgebra approach to study the properties of the given expression.

Let $A = {\Bbb R} [x, y, z]$. Then, we can define a superalgebra $A^{\vee}$ as follows:

$$A^{\vee} = \{f \in A \mid f \text{ is a polynomial in } x, y, z \text{ with real}\}$$

Q: How do we prove the existence of four polynomials $p_1, p_2, p_3, p_4$?

A: To prove the existence of four polynomials $p_1, p_2, p_3, p_4$, we can use the following steps:

1. Step 1: Simplify the given expression using the substitution method.
2. Step 2: Use the sum of squares method to express the simplified expression as the sum of squares of four polynomials.
3. Step 3: Verify that the four polynomials obtained in Step 2 satisfy the required conditions.

Q: What are the required conditions for the four polynomials $p_1, p_2, p_3, p_4$?

A: The required conditions for the four polynomials $p_1, p_2, p_3, p_4$ are that they are indeed polynomials in $x, y, z$ with real coefficients.

Q: What are some applications of the sum of squares method?

A: The sum of squares method has numerous applications in various fields, including optimization, control theory, and machine learning.

Q: What is the significance of the superalgebra approach?

A: The superalgebra approach is a method used to study the properties of polynomials and their relationships. It is a powerful tool in real algebraic geometry and has numerous applications in various fields.

Q: How does the problem relate to real algebraic geometry?

A: The problem is a classic example of the application of the sum of squares method in real algebraic geometry. It involves the study of the properties of polynomials and their relationships using algebraic and geometric methods.

Q: What are some related problems or results?

A: Some related problems or results include:

• Theorem 1: Let $f(x) \in {\Bbb R} [x]$ be a polynomial. Then, $f(x) \geq 0$ for all $x \in {\Bbb R}$ if and only if there exist polynomials $g_1, g_2, \ldots, g_n \in {\Bbb R} [x]$ such that

$$f(x) = g_1^2(x) + g_2^2(x) + \cdots + g_n^2(x)$$

• Theorem 2: Let $A = {\Bbb R} [x, y, z]$. Then, we can define a superalgebra $A^{\vee}$ as follows:

$$A^{\vee} = \{f \in A \mid f \text{ is a polynomial in } x, y, z \text{ with real}\}$$

Q: What are some open problems or research directions?

A: Some open problems or research directions include:

• Open Problem 1: Prove that there exist four polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ in $x, y, z$ so that

$$\left ( x^{2}+ y^{2}+ z^{2} \right )^{4}- 8\left ( z^{4}x^{4}+ x^{4}y^{4}+ y^{4}z^{4} \right ) = p_{1}^{2}+ p_{2}^{2}+ p_{3}^{2}+ p_{4}^{2}$$

• Open Problem 2: Study the properties of the superalgebra $A^{\vee}$ and its applications in real algebraic geometry.