cmr a 2 b 2 c 2 ab bc ca

$$q := a^2 + b^2 + c^2 - ab - bc - ca = \frac12 \begin{bmatrix} a\\ b \\ c\end{bmatrix}^\top \underbrace{\begin{bmatrix} 2 & -1 & -1\\ -1 & 2 & -1 \\ -1 & -1 & 2\end{bmatrix}}_{=: {\rm L}} \begin{bmatrix} a\\ b \\ c\end{bmatrix}$$

where matrix $\rm L$ is the Laplacian of the cycle graph with $3$ vertices, whose (signed) incidence matrix is

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$${\rm C} = \begin{bmatrix} -1 & \color{blue}{1} & 0\\ 0 & -1 & \color{blue}{1} \\ \color{blue}{1} & 0 & -1\end{bmatrix}$$

Since $\rm L = C^\top C$, we obtain the following sum of squares (SOS) decomposition

$$2q = (\color{blue}{b} - a)^2 + (\color{blue}{c} - b)^2 + (\color{blue}{a} - c)^2 \geq 0$$

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which is the SOS decomposition proposed by Mark Bennet. Since matrix $\rm L$ is rank-$2$, a terser SOS decomposition with only $2$ terms can easily be found — say, via the Cholesky decomposition.

Using Macaulay2,

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Macaulay2, version 1.16
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, MinimalPrimes, PrimaryDecomposition, ReesAlgebra, TangentCone, Truncations

i1 : needsPackage( "SumsOfSquares" );
--loading configuration for package "NumericalAlgebraicGeometry" from tệp tin /Users/rodrigo/Library/Application Support/Macaulay2/init-NumericalAlgebraicGeometry.m2
--loading configuration for package "Bertini" from tệp tin /Users/rodrigo/Library/Application Support/Macaulay2/init-Bertini.m2
--warning: symbol "Verbosity" in MinimalPrimes.Dictionary is shadowed by a symbol in SemidefiniteProgramming.Dictionary
--  use the synonym MinimalPrimes$Verbosity

i2 : R = QQ[a,b,c];

i3 : q = a^2 + b^2 + c^2 - a*b - a*c - b*c 

      2          2                2
o3 = a  - a*b + b  - a*c - b*c + c

o3 : R

i4 : sosPoly solveSOS q

             1    1  2    3        2
o4 = (1)(a - -b - -c)  + (-)(b - c)
             2    2       4

o4 : SOSPoly

i5 : tex o4

o5 = $\texttt{SOSPoly}\left\{\texttt{coefficients}\,\Rightarrow\,\left\{1,\,\frac{3}{4}\right\},\,\texttt{generators}\,\Rightarrow\,\left\{a-\frac{1}{2}\,b-\frac{1}{2}\,c,\,b-c\right
     \},\,\texttt{ring}\,\Rightarrow\,R\right\}$

In $\TeX$,

$$\texttt{SOSPoly}\left\{\texttt{coefficients}\,\Rightarrow\,\left\{1,\,\frac{3}{4}\right\},\,\texttt{generators}\,\Rightarrow\,\left\{a-\frac{1}{2}\,b-\frac{1}{2}\,c,\,b-c\right\},\,\texttt{ring}\,\Rightarrow\,R\right\}$$

polynomials sum-of-squares-method macaulay2 graph-laplacian