Questions tagged [sum-of-squares-method]

Proofs of inequalities by the Sum of Squares method (SOS).

288 questions
Filter by
Sorted by
Tagged with
1 vote
65 views

Writing $1-xyz$ as a sum of squares

Can you write $1 - xyz$ in the form $p + q (1 - x^{2}-y^{2}-z^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z]$? For instance, in ...
1 vote
113 views

Decomposing $\sum_{i = 0}^{2n} x^i$ as a simple sum of squares

As we have $\sum_{i = 0}^{2n} x^i = (x^{2n + 1} - 1) / (x - 1)$, the polynomial is positive. So we know that there is a decomposition as a sum of squares. Is there a closed simple form for such a ...
• 23
152 views

Find parameters $a,b$ such that $x^6-2 x^5+2 x^4+2 x^3-x^2-2 x+1-\left(x^3-x^2+a x+b\right)^2>0$

The probrem is to prove that $$x^6-2 x^5+2 x^4+2 x^3-x^2-2 x+1>0.$$ (the minimum value is about 0.02, tested by wolframalpha.) I use sos(sum of squares) method, my idea is to reduce the degree of ...
• 119
56 views

Sum of Two Squares of a Quartic

I am trying to write a quartic in sum of two squares, can anyone help me with the following polynomial: $$6y_0^{4}+6y_0^{2}y_1^{2}+y_1^4+4y_0y_1^{2}y_2+4y_0^{2}y_2^{2}+ 6y_1^{2}y_2^2+6y_2^{4},$$ one ...
• 401
64 views

• 409
159 views

Nice problem: Prove that: $ab+bc+ca \ge \sum{\sqrt{a^2+b^2+3}}$

Problem: Let $a,b,c>0:a+b+c=abc.$ Prove that: $$ab+bc+ca\ge \sqrt{a^2+b^2+3}+\sqrt{b^2+c^2+3}+\sqrt{c^2+a^2+3}$$ Please help me give a hint to get a nice proof! My attempts after squaring both side,...
• 409
1 vote
109 views

Find another sum of squares for $3^{12}-6^6+2^{12}$

I have a question about factorization of number $3^{12}-6^6+2^{12}$. By completing the square one can show that$$3^{12}-6^6+2^{12} = (3^6-2^6)^2+6^6 = 665^2+216^2$$ If we can find another ...
• 704
100 views

123 views

Prove $\sum_{cyc}\frac{xy+1}{(x+y)^2}\geq 3$ when $x^2+y^2+z^2+(x+y+z)^2\leq 4$.

Let $x,y,z\in \Bbb{R}^+$ such that $x^2+y^2+z^2+(x+y+z)^2\leq 4$. Prove that $$\sum_{cyc}\frac{xy+1}{(x+y)^2}\geq 3.$$ As there are three fractions in the left side and a single term in the right ...
• 3,823
93 views

show this inequality $\sum_{cyc}\sqrt{a(b+c)}(b^2+c^2-a^2-bc)\ge 0$

let $a,b,c>0$,show this inequality $$\sqrt{a(b+c)}(b^2+c^2-a^2-bc)+\sqrt{b(c+a)}(c^2+a^2-b^2-ca)+\sqrt{c(a+b)}(a^2+b^2-c^2-ab)\ge 0$$ I want use S-O-S methods to solve ,But I can't, see this ...
147 views

• 130k
274 views

Find $g(x,y,z);p(x,y,z)\ge 0$ so that $f(x,y,z):=x\cdot g(x,y,z)+p(x,y,z)$

We have the following fact: (I don't remember where I read it, but there is.) If $f(x)$ is which is non-negative for $x\ge 0,$ then $f(x)=g(x)+x\cdot h(x),$ where $g(x)$ and $h(x)$ are SOS. So I ...
• 1,950
412 views

• 22.6k
65 views

• 9,572
1 vote
34 views

If the real zeroes of real polynomial $p(x,y)$ are disjoint points and curves, is $p(x,y)$ a positive sum of squares?

For example, $p(x,y) = x^2(x-1)^2 + y^2(y-1)^2$ has real zeroes in the set $\{(0,0), (0, 1), (1, 0), (1, 1)\}$ and admits a decomposition into a sum of squares. How can I find decompositions like this ...
• 297
1 vote