Math 2.0

Field theory and polynomials Real fields and ordered fields Problem

Hilbert's seventeenth problem

Builds on Real numbersPolynomial ringRational functionFormally real fieldReal closure

Problem

Let n≥1n \ge 1 and let f∈ℝ[x1,…,xn]f \in \R[x_1, \dots, x_n] be a polynomial with real coefficients such that f(a)≥0f(a) \ge 0 for every a∈ℝna \in \R^n. Is ff a sum of squares of rational functions, that is, are there g1,…,gm∈ℝ(x1,…,xn)g_1, \dots, g_m \in \R(x_1, \dots, x_n) with f=g12+⋯+gm2f = g_1^2 + \cdots + g_m^2?

Remarks

Emil Artin answered the question positively in 1927. His proof uses the theory of formally real fields and real closures. If ff were not a sum of squares, some ordering of ℝ(x1,…,xn)\R(x_1, \dots, x_n) would make ff negative. Passing to a real closure then yields a real point where f<0f < 0. Rational functions are needed: a nonnegative polynomial need not be a sum of squares of polynomials. An example is the Motzkin polynomial x4y2+x2y4−3x2y2+1x^4 y^2 + x^2 y^4 - 3x^2 y^2 + 1, which is not a sum of squares of polynomials. It is nonnegative by the inequality of arithmetic and geometric means applied to x4y2x^4 y^2, x2y4x^2 y^4 and 11. Hilbert showed that every nonnegative polynomial is a sum of squares of polynomials exactly in these cases: one variable, degree 22, or two variables and degree 44. Pfister showed that 2n2^n squares of rational functions always suffice.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Formally real field, concept, no evidence levelFormally real fieldconceptGroup, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionReal closure, concept, no evidence levelReal closureconceptReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionRing, definition, no evidence levelRingdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…statementRational function, definition, no evidence levelRational functiondefinitionHilbert's seventeenth problem, problem, no evidence levelHilbert's seventeenth…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Formally real field concept
  2. Group concept
  3. Limit point of a set definition
  4. Real closure concept
  5. Real numbers object
  6. Sequence definition
  7. Ring definition
  8. Limit of a function definition
  9. Real sequence definition
  10. Polynomial ring definition
  11. Continuity at a point definition
  12. Bounded sequence definition
  13. Limit of a sequence definition
  14. Continuous function definition
  15. Epsilon-delta definition of continuity statement
  16. Convergent sequence definition
  17. Epsilon-N definition of the limit of a sequence statement
  18. Sequential continuity definition
  19. Continuity is equivalent to sequential continuity statement
  20. Convergent sequences are bounded statement
  21. Limit of a product of sequences statement
  22. Product of continuous functions is continuous statement
  23. Limit of a quotient of sequences statement
  24. Polynomial functions are continuous statement
  25. Quotient of continuous functions is continuous statement
  26. Rational function definition
  27. Hilbert's seventeenth problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_seventeenth_problem
About this page
TagJ45F4F13VB
URIm2://J45F4F13VB
Kindproblem
WritingW0 stub
MSC12D15, 11E25, 14P05
StatusThis copy of the site has no build data. Run m2 status J45F4F13VB.
AboutFormally real field, Real closure
UsesReal numbers, Polynomial ring, Rational function, Formally real field, Real closure
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Changed2 November 2026

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