Mathematics

Questions tagged [sum-of-squares-method]

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

Let $f(a,b,c)=S_a(b-c)^2+S_b(a-c)^2+S_c(a-b)^2$. Then:

  • If $S_a\geq0$, $S_b\geq0$ and $S_c\geq0$ then $f(a,b,c)\geq0$.

  • If $S_a+S_b+S_c\geq0$ and $S_aS_b+S_aS_c+S_bS_c\geq0$ then $f(a,b,c)\geq0$.

  • If $a\geq b\geq c$, $S_a\geq0$, $S_b\geq0$ and $S_b+S_c\geq0$ then $f(a,b,c)\geq0$.

  • If $a\geq b\geq c$, $S_c\geq0$, $S_b\geq0$ and $S_b+S_a\geq0$ then $f(a,b,c)\geq0$.

  • If $a\geq b\geq c$, $S_c\geq0$, $S_b\geq0$ and $a^2S_b+b^2S_a\geq0$ then $f(a,b,c)\geq0$.

336 questions
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Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

Let $a$ , $b$ and $c$ be non-negative numbers. Prove that: $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}.$$ I have a proof, but my proof is very ugly: it's enough to prove a polynomial inequality of degree $15$ .…
Michael Rozenberg
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4 answers

Proving $abcd+3\geq a+b+c+d$

If $a,b,c,d$ are non negatives and $a^2+b^2+c^2+d^2=3$ prove that $$abcd+3\ge a+b+c+d$$ The inequality is not as simple as it looks.The interesting part is that the equality occurs when $a=0,b=c=d=1$ upto permutation .(I don't know if there are…
Hari Ramakrishnan Sudhakar
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Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$

Let $x_1,x_2,\ldots,x_n > 0$ such that $\dfrac{1}{1+x_1}+\cdots+\dfrac{1}{1+x_n}=1$ . Prove the following inequality. $$\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left…
Jacob Willis
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Write $(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$ as a sum of (three) squares of quadratic forms

The quartic form $$(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$$ is non-negative for all real $x$, $y$, $z$, as one can check (with some effort). A theorem of Hilbert implies that there exist quadratic forms $Q_1$, $Q_2$, $Q_3$ so that $$(x^2 +…
orangeskid
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Given $P(x)=x^{4}-4x^{3}+12x^{2}-24x+24,$ then $P(x)=|P(x)|$ for all real $x$

$$P(x)=x^{4}-4x^{3}+12x^{2}-24x+24$$ I'd like to prove that $P(x)=|P(x)|$ . I don't know where to begin. What would be the first step?
lmc
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Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$

I am currently attempting to prove the following inequality $\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b} \geq \dfrac{3}{2}$ for all $ a,b,c>0$ My instinctive plan of attack is to use the AM/GM inequality with $x_1=\dfrac{a}{b+c}$ etc. Using that I…
Trogdor
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On the properties of sum-of-squares polynomials

Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$ , then $f$ is SOS. Definition 2. If an $n$ -variate polynomial $f$ is nonnegative on $\Bbb R^n$ , then $f$ is…
khashayar
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Find three polynomials whose squares sum up to $x^4 + y^4 + x^2 + y^2$

Prove that $$p(x,y) = x^4 + y^4 + x^2 + y^2$$ can be written as a sum of squares of three polynomials over $x,y$ for real numbers.
user31869
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5 answers

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Let $a,b,c>0$ how to prove that : $$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$ I find that $$\ \frac{ab}{a^{2}+3b^{2}}=\frac{1}{\frac{a^{2}+3b^{2}}{ab}}=\frac{1}{\frac{a}{b}+\frac{3b}{a}} $$ By AM-GM $$\…
Frank
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Express $x^4 + y^4 + x^2 + y^2$ as sum of squares of three polynomials in $x,y$

I don't know any identity that'd help me simplify it. I know of Brahmagupta's identity and tried using but no good. Any hints? Edit: So far I've tried various things, $(x^2+1/2)^2 + (y^2 +1/2)^2 -1/2$ This doesn't get me anywhere. I tried solving in…
Sajid Rizvi
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Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
the_random_guy42
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SOS: Proof of the AM-GM inequality

"A basic strategy in tackling inequalities of few variables is to write things into sum of squares..." is a quote from an answer, intended as commenting the post entitled "Can this inequality proof be demystified?". I'd like to know what the S um O…
Hanno
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If $2(bc^2+ca^2+ab^2) = b^2c+c^2a+a^2b+3abc, $ then prove that triangle $ABC$ is equilateral

In a triangle $ABC,$ if $2(bc^2+ca^2+ab^2) = b^2c+c^2a+a^2b+3abc, $ then prove that triangle $ABC$ is equilateral $\displaystyle \frac{2(bc^2+ca^2+ab^2)}{abc} = \frac{b^2c+c^2a+a^2b+3abc}{abc}$ $\displaystyle…
DXT
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A positive polynomial of Schur as a sum of squares of polynomials

This is a follow-up this question on MathOverflow. The previous time I asked about SOS was Choi Lam homogeneous polynomials as sums of squares . With positive integer $k,$ and real variables $x,y,z$ that are allowed to take on positive or negative…
Will Jagy
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Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$ . Definition: A polynomial $f$ is $d$ -SOS if there exist $g_{1}, \dots, g_{k} \in…
Tio Miserias
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