[Polynomials] A Case of a Non-Negative Polynomial that cannot be written as a Sum of Squares

Let's show that the Motzkin polynomial cannot be expressed as the sum of squares. Assume for the sake of contradiction, that \[M(x,y) = x^4y^2+x^2y^4 +1-3x^2y^2 = \sum_{j=1}^{p} f_j (x,y)^2 \]\[=\sum (A_j+B_jx+C_jy+D_jx^2+E_jxy+F_jy^2+G_jx^3+H_jx^2y+I_jxy^2+J_jy^3)^2,\]

where \(A_j,B_j,..., I_j, J_j\) are all real constants. By comparing the above expression to Motzkin's polynomial, we see that some of the coefficients in \(\sum f_j (x,y)^2\) equal zero, because Motzkin's polynomial does not contain some terms in \(\sum f_j(x,y)^2.\)

Firstly, Motzkin's polynomial does not have \(x^6, y^6\), so we can eliminate the constants \(G_j\) and \(J_j\), that is, \(G_j=J_j=0\).

So now \[f_j(x,y)= A_j+ B_jx + C_jy + D_jx^2 + E_jxy + F_jy^2 + H_jx^2y + I_jxy^2.\]

Additionally, Motzkin's polynomial does not have the terms \(x^4, y^4\), so we know that \(D_j=F_j=0\).

So now \[f_j(x,y)= A_j + B_jx + C_jy + E_jxy + H_jx^2y + I_jxy^2.\]

Likewise, there is no \(x^2\) term in the Motzkin polynomial, which means that \(\sum B_j^2x^2=0\), hence each \(B_j=0\). Finally,there is no \(y^2\) term, so the coefficient of \(y^2\) of \(\sum f_j(x,y)^2 \) is zero, so \(\sum C_j^2 =0\) and each \( C_j=0.\)

Therefore, we are left with \[f_j(x,y)= A_j + E_jxy + H_jx^2y + I_jxy^2,\] where \(A_j, E_j, H_j, I_j\) are constants.

Following this procedure, we see that the coefficient of the \(x^2y^2\) term in the Motzkin polynomial is -3. This means that

\[\sum 2F_jD_j+E_j^2+2B_jI_j+2C_jH_j = \sum (2F_jD_j+E_j^2)^2 = -3.\]

But if we take a look at \(D_j\), we see that there is no \(x^4\) term in the Motzkin polynomial, so \(\sum D_j^2+2B_jG_j=0\). But we found above that \(B_j=0\), so \(\sum D_j^2=0\), hence each \(D_j=0\). Therefore, \[\sum (2F_jD_j+E_j^2) = \sum E_j^2 =-3.\]

Since the sum of all the \(E_j^2\) terms cannot be negative, this is a contradiction.

Hence we conclude that the Motzkin polynomial, \(M(x,y) = x^4y^2+x^2y^4 +1-3x^2y^2\), is an example of a non-negative polynomial of two variables that cannot be written as the sum of squares.

Previously, we showed that the Motzkin polynomial is non-negative using the AM-GM inequality. This polynomial caught our interest, because it is non-negative in \(\mathbb{R}^2\), but cannot be expressed as the sum of two squares of polynomials. The Motzkin polynomial illustrates that although a non-negative polynomial in one variable can always be expressed as the sum of squares, this is not necessarily the case for a polynomial in two variables.

Note by Tasha Kim
3 months, 1 week ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...