HILBERT'S 23 PROBLEMS
Paris, 1900
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SOLVEDPARTIALLY SOLVEDOPENTOO VAGUE
#17
SOLVED
Expression of Definite Forms as Sums of Squares
Artin (1927)
Can every positive semidefinite rational function be written as a sum of squares of rational functions?
HISTORY
Artin (1927) proved yes. This is Artin's solution to Hilbert's 17th problem, using the theory of ordered fields.
Founded real algebraic geometry. The constructive aspects (how many squares?) led to ongoing research in optimization and semidefinite programming.
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