  Brascamp-Lieb inequality is an important mathematical tool in analysis,
geometry and information theory. There are various ways to prove Brascamp-Lieb
inequality such as heat flow method, Brownian motion and subadditivity of the
entropy. While Brascamp-Lieb inequality is originally stated in Euclidean
Space, discussed Brascamp-Lieb inequality for discrete Abelian group and
discussed Brascamp-Lieb inequality for Markov semigroups.
  Many mathematical inequalities can be formulated as algebraic inequalities
which asserts some given polynomial is nonnegative. In 1927, Artin proved that
any non- negative polynomial can be represented as a sum of squares of rational
functions, which can be further formulated as a polynomial certificate of the
nonnegativity of the polynomial. This is a Sum of Square proof of the
inequality. Take the degree of the polynomial certificate as the degree of Sum
of Square proof. The degree of an Sum of Square proof determines the complexity
of generating such proof by Sum of Square algorithm which is a powerful tool
for optimization and computer aided proof.
  In this paper, we give a Sum of Square proof for some special settings of
Brascamp- Lieb inequality following and and discuss some applications of
Brascamp-Lieb inequality on Abelian group and Euclidean Sphere. If the original
description of the inequality has constant degree and d is constant, the degree
of the proof is also constant. Therefore, low degree sum of square algorithm
can fully capture the power of low degree finite Brascamp-Lieb inequality.
