PolynomialSumOfSquaresList

PolynomialSumOfSquaresList[f,vars]

attempts to find polynomials with real coefficients {f1,,fn} such that f==f12++fn2.

Details

  • PolynomialSumOfSquaresList is typically used to show that a real polynomial is non-negative.
  • f should be a polynomial in vars with real numeric coefficients.

Examples

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Basic Examples  (1)

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

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

Scope  (6)

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:

Applications  (1)

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:

Properties & Relations  (3)

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 :

Possible Issues  (1)

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

Wolfram Research (13), PolynomialSumOfSquaresList, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialSumOfSquaresList.html.

Text

Wolfram Research (13), PolynomialSumOfSquaresList, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialSumOfSquaresList.html.

CMS

Wolfram Language. 13. "PolynomialSumOfSquaresList." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PolynomialSumOfSquaresList.html.

APA

Wolfram Language. (13). PolynomialSumOfSquaresList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialSumOfSquaresList.html

BibTeX

@misc{reference.wolfram_2021_polynomialsumofsquareslist, author="Wolfram Research", title="{PolynomialSumOfSquaresList}", year="13", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialSumOfSquaresList.html}", note=[Accessed: 24-January-2022 ]}

BibLaTeX

@online{reference.wolfram_2021_polynomialsumofsquareslist, organization={Wolfram Research}, title={PolynomialSumOfSquaresList}, year={13}, url={https://reference.wolfram.com/language/ref/PolynomialSumOfSquaresList.html}, note=[Accessed: 24-January-2022 ]}