Find a sum-of-squares representation of a polynomial:

A polynomial representable as a sum of squares is non-negative:

Find a sum-of-squares representation of a polynomial:

The polynomial is non-negative:

Find a sum-of-squares representation of a polynomial with exact real numeric coefficients:

Find a sum-of-squares representation of a polynomial with machine-precision real coefficients:

The difference of and the sum of squares has small coefficients:

Find a sum-of-squares representation of a polynomial with arbitrary-precision real coefficients:

The difference of and the sum of squares has arbitrary-precision zero coefficients:

A polynomial may not be representable as a sum of squares:

The polynomial attains negative values, hence it cannot be a sum of squares:

PolynomialSumOfSquaresList may fail even if a sum-of-squares representation exists:

Prove non-negativity of a large polynomial:

Compute a sum-of-squares representation:

is a sum of squares, hence it is non-negative:

FindInstance uses a method based on sum-of-squares representation here:

Deciding non-negativity of f using CylindricalDecomposition takes much more time:

A polynomial that has a sum-of-squares representation is non-negative:

Use FindInstance to show that is non-negative:

The Motzkin polynomial is non-negative, but is not a sum of squares:

Use Resolve to show that is non-negative:

Use MinValue to find the infimum of values of a polynomial:

PolynomialSumOfSquaresList fails for the non-negative polynomial :

PolynomialSumOfSquaresList succeeds for the strictly positive polynomial :

PolynomialSumOfSquaresList may fail even if a sum-of-squares representation exists: