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	~~'''Hilbert's seventeenth problem'''~~ is ~~one of the 23 [[Hilbert problems]] set out in a celebrated list compiled in 1900~~ by ~~[[David Hilbert]]. It concerns~~ the ~~expression~~ of ~~[[definite]] [[rational function]]s as [[sum]]s of [[quotient]]s of [[Square (algebra)\|square]]s. The original question may be stated as:~~		The writer is known by the name of Figures Wunder. For years he's been living in North Dakota and his family members loves it. Hiring is her day occupation now but she's usually needed her own business. To do aerobics is a factor that I'm totally addicted to.<br><br>Feel free to visit my page :: [http://www.blaze16.com/blog/255682 http://www.blaze16.com/]

	* Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?

	This was solved in the affirmative, in 1927, by [[Emil Artin]], for positive definite functions over the reals or more generally [[real-closed field]]s. An algorithmic solution was found by [[Charles Neal Delzell\|Charles Delzell]] in 1984.<ref>{{cite journal \| zbl=0547.~~12017 \| last=Delzell \| first=C.N. \| title=A continuous, constructive solution to Hilbert~~'s 17th problem \| journal=[[Inventiones Mathematicae]] \| volume=76 \| pages=365–384 \| year=1984 \| doi=10.1007/BF01388465 \| url=http://www.springerlink.com/content/m18152477392j1t2/ }}</ref> A result of [[Albrecht Pfister (mathematician)\|Albrecht Pfister]]<ref name=Pf1967>{{cite journal \| last=Pfister \| first=Albrecht \| authorlink=Albrecht Pfister (mathematician) \| title=Zur Darstellung definiter Funktionen als Summe von Quadraten \| language=German \| journal=[[Inventiones Mathematicae]] \| volume=4 \| pages=229–237 \| year=1967 \| zbl=0222.10022 }}</ref> shows that a positive semidefinite form in ~~''n'' variables can be expressed as a sum of 2<sup>''n''</sup> squares~~.~~<ref name=Lam391>Lam (2005) p.391</ref>~~

	~~Dubois showed in 1967 that the answer~~ is ~~negative in general for [[ordered field]]s.<ref>{{cite journal \| zbl=0164.04502 \| last=Dubois \| first=D.W. \| title=Note on Artin's solution of Hilbert~~'s ~~17th problem \| journal=Bull. Am. Math. Soc~~. ~~\| volume=73 \| pages=540–541 \| year=1967 }}</ref> In this case one can say that a positive polynomial~~ is a ~~sum of weighted squares of rational functions with positive coefficients.<ref>Lorenz (2008) p.16</ref>~~

	~~A generalization to the matrix case (matrices with rational function entries~~ that are always positive semidefinite are sums of symmetric squares) was given by Gondard, [[Paulo Ribenboim\|Ribenboim]]<ref>{{cite journal \| zbl=0298.12104 \| mr=432613 \| last1=Gondard \| first1=Danielle \| last2=Ribenboim \| first2=Paulo \| author2-link=Paulo Ribenboim \| title=Le 17e problème de Hilbert pour les matrices \| journal=Bull. Sci. Math. (2) \| volume=98 \| year=1974 \| number=1 \| pages=49–56 }}</ref> and Procesi, Schacher,<ref>{{cite journal \| mr=432612 \| zbl=0347.16010 \| last1=Procesi \| first1=Claudio \| last2=Schacher \| first2=Murray \| title=A non-commutative real Nullstellensatz and Hilbert'~~s 17th problem \| journal=Ann. of Math~~. ~~(2) \| volume=104\| year=1976 \| number=3 \| pages=395–406 }}~~<~~/ref~~> ~~with an elementary proof given by Hillar and Nie.~~<~~ref~~>~~{{cite journal \| zbl=1126.12001 \| last1=Hillar \| first1=Christopher J. \| last1=Nie \| first1=Jiawang \| title=An elementary and constructive solution~~ to ~~Hilbert's 17th problem for matrices \| journal=Proc. Am. Math. Soc. \| volume=136 \| number=1 \| pages=73–76 \| year=2008 \| arxiv=math/0610388 }}</ref>~~

	~~The formulation of the question takes into account that there are polynomials, for example<ref>[[Marie-Françoise Roy]]. The role of Hilbert's problems in real algebraic geometry.~~
	~~Proceedings of the ninth EWM Meeting, Loccum, Germany 1999</ref>~~

	:~~<math>f(x,y,z)=z^6+x^4y^2+x^2y^4-3x^2y^2z^2 \, </math>~~

	~~which are non-negative over reals and yet which cannot be represented as a sum of squares of other polynomials, as Hilbert had shown in 1888 but without giving an example~~: ~~the first explicit example was found by Motzkin in 1966.~~

	Explicit sufficient conditions for a polynomial to be a sum of squares of other polynomials have been found <ref>{{cite journal \| last=Lasserre \| first=Jean B. \| title=Sufficient conditions for a real polynomial to be a sum of squares \| journal=[[~~Arch. Math.]] \| volume=89 \| number=5 \| pages=390–398 \| year=2007 \| zbl=1149.11018 \| url=~~http://www.~~optimization-online~~.~~org/DB_HTML/2007/02~~/~~1587.html }}<~~/~~ref><ref>[~~http://www.~~mathcs~~.emory.edu/~vicki/pub/sos.pdf]</ref> However every real nonnegative polynomial can be approximated as closely as desired (in the <math>l_1</math>-norm of its coefficient vector) by a sequence of polynomials that are sums of squares of polynomials.<ref>{{cite journal \| last=Lasserre \| first=Jean B. \| title=A sum of squares approximation of nonnegative polynomials \| journal=SIAM Rev. \| volume=49 \| number=4 \| pages=651–669 \| year=2007 \| issn=0036-1445 \| zbl=1129.12004 \| url=http://portal.acm.org/~~citation.cfm?id=1330215.1330223&coll=GUIDE&dl= }}</ref>~~

	~~It is an open question what is the smallest number~~

	~~:<math>v(n,d), \, </math>~~

	~~such that any ''n''-variate, non-negative polynomial of degree ''d'' can be written as sum of at most <math>v(n,d)</math> square rational functions over the reals.~~

	~~The best known result ({{As of\|2008\|lc=on}}) is~~

	~~:<math>v(n,d)\leq2^n, \, </math>~~

	~~due to Pfister in 1967.<ref name=Pf1967/>~~

	In complex analysis the Hermitian analogue, requiring the squares to be squared norms of holomorphic mappings, is somewhat more complicated, but true for positive polynomials by a result of Quillen.<ref>{{cite journal \| zbl=0198.35205 \| last1=Quillen \| first1=Daniel G. \| title=On the representation of hermitian forms as sums of squares \| journal=Invent. Math. \| volume=5 \| pages=237–242 \| year=1968 }}</ref> The result of Pfister on the other hand fails in the Hermitian case, that is there is no bound on the number of squares required, see D'Angelo–Lebl.<ref>{{cite journal \| zbl=06028329 \| last1=D'Angelo \| first1=John P. \| last2=Lebl \| first2=Jiri \| title=Pfister's theorem fails in the Hermitian case \| journal=Proc. Am. Math. Soc. \| volume=140 \| number=4 \| pages=1151–1157 \| year=2012 \| arxiv=1010.3215 }}</ref>

	~~==See also==~~
	* [[Polynomial SOS]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book \| editor=Felix E. Browder \| editor-link=Felix Browder \| title=Mathematical Developments Arising from Hilbert Problems \| series=[[Proceedings of Symposia in Pure Mathematics]] \| volume=XXVIII.2 \| year=1976 \| publisher=[[American Mathematical Society]] \| isbn=0-8218-1428-1 \| first=Albrecht \| last=Pfister \| authorlink=Albrecht Pfister (mathematician) \| chapter=Hilbert's seventeenth problem and related problems on definite forms \| pages=483–489 }}
	* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 }}
	* {{cite book \| first=Falko \| last=Lorenz \| title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics \| year=2008 \| publisher=[[Springer-Verlag]] \| isbn=978-0-387-72487-4 \| pages=15–27 \| zbl=1130.12001 }}
	* {{cite book \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. R. \| last=Rajwade \| publisher=[[Cambridge University Press]] \| year=1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}

	~~{{Hilbert's problems}}~~

	~~{{DEFAULTSORT:Hilbert's Seventeenth Problem}}~~
	~~[[Category:Real algebraic geometry]]~~
	~~[[Category:Hilbert's problems\|#17]~~]

en>Lockley: replace context tag with technical tag

2012-06-19T14:22:51Z

replace context tag with technical tag

New page

'''Hilbert's seventeenth problem''' is one of the 23 [[Hilbert problems]] set out in a celebrated list compiled in 1900 by [[David Hilbert]]. It concerns the expression of [[definite]] [[rational function]]s as [[sum]]s of [[quotient]]s of [[Square (algebra)|square]]s. The original question may be stated as:

* Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?

This was solved in the affirmative, in 1927, by [[Emil Artin]], for positive definite functions over the reals or more generally [[real-closed field]]s. An algorithmic solution was found by [[Charles Neal Delzell|Charles Delzell]] in 1984.<ref>{{cite journal | zbl=0547.12017 | last=Delzell | first=C.N. | title=A continuous, constructive solution to Hilbert's 17th problem | journal=[[Inventiones Mathematicae]] | volume=76 | pages=365–384 | year=1984 | doi=10.1007/BF01388465 | url=http://www.springerlink.com/content/m18152477392j1t2/ }}</ref> A result of [[Albrecht Pfister (mathematician)|Albrecht Pfister]]<ref name=Pf1967>{{cite journal | last=Pfister | first=Albrecht | authorlink=Albrecht Pfister (mathematician) | title=Zur Darstellung definiter Funktionen als Summe von Quadraten | language=German | journal=[[Inventiones Mathematicae]] | volume=4 | pages=229–237 | year=1967 | zbl=0222.10022 }}</ref> shows that a positive semidefinite form in ''n'' variables can be expressed as a sum of 2<sup>''n''</sup> squares.<ref name=Lam391>Lam (2005) p.391</ref>

Dubois showed in 1967 that the answer is negative in general for [[ordered field]]s.<ref>{{cite journal | zbl=0164.04502 | last=Dubois | first=D.W. | title=Note on Artin's solution of Hilbert's 17th problem | journal=Bull. Am. Math. Soc. | volume=73 | pages=540–541 | year=1967 }}</ref> In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive coefficients.<ref>Lorenz (2008) p.16</ref>

A generalization to the matrix case (matrices with rational function entries that are always positive semidefinite are sums of symmetric squares) was given by Gondard, [[Paulo Ribenboim|Ribenboim]]<ref>{{cite journal | zbl=0298.12104 | mr=432613 | last1=Gondard | first1=Danielle | last2=Ribenboim | first2=Paulo | author2-link=Paulo Ribenboim | title=Le 17e problème de Hilbert pour les matrices | journal=Bull. Sci. Math. (2) | volume=98 | year=1974 | number=1 | pages=49–56 }}</ref> and Procesi, Schacher,<ref>{{cite journal | mr=432612 | zbl=0347.16010 | last1=Procesi | first1=Claudio | last2=Schacher | first2=Murray | title=A non-commutative real Nullstellensatz and Hilbert's 17th problem | journal=Ann. of Math. (2) | volume=104| year=1976 | number=3 | pages=395–406 }}</ref> with an elementary proof given by Hillar and Nie.<ref>{{cite journal | zbl=1126.12001 | last1=Hillar | first1=Christopher J. | last1=Nie | first1=Jiawang | title=An elementary and constructive solution to Hilbert's 17th problem for matrices | journal=Proc. Am. Math. Soc. | volume=136 | number=1 | pages=73–76 | year=2008 | arxiv=math/0610388 }}</ref>

The formulation of the question takes into account that there are polynomials, for example<ref>[[Marie-Françoise Roy]]. The role of Hilbert's problems in real algebraic geometry.
Proceedings of the ninth EWM Meeting, Loccum, Germany 1999</ref>

:<math>f(x,y,z)=z^6+x^4y^2+x^2y^4-3x^2y^2z^2 \, </math>

which are non-negative over reals and yet which cannot be represented as a sum of squares of other polynomials, as Hilbert had shown in 1888 but without giving an example: the first explicit example was found by Motzkin in 1966.

Explicit sufficient conditions for a polynomial to be a sum of squares of other polynomials have been found <ref>{{cite journal | last=Lasserre | first=Jean B. | title=Sufficient conditions for a real polynomial to be a sum of squares | journal=[[Arch. Math.]] | volume=89 | number=5 | pages=390–398 | year=2007 | zbl=1149.11018 | url=http://www.optimization-online.org/DB_HTML/2007/02/1587.html }}</ref><ref>[http://www.mathcs.emory.edu/~vicki/pub/sos.pdf]</ref> However every real nonnegative polynomial can be approximated as closely as desired (in the <math>l_1</math>-norm of its coefficient vector) by a sequence of polynomials that are sums of squares of polynomials.<ref>{{cite journal | last=Lasserre | first=Jean B. | title=A sum of squares approximation of nonnegative polynomials | journal=SIAM Rev. | volume=49 | number=4 | pages=651–669 | year=2007 | issn=0036-1445 | zbl=1129.12004 | url=http://portal.acm.org/citation.cfm?id=1330215.1330223&coll=GUIDE&dl= }}</ref>

It is an open question what is the smallest number

:<math>v(n,d), \, </math>

such that any ''n''-variate, non-negative polynomial of degree ''d'' can be written as sum of at most <math>v(n,d)</math> square rational functions over the reals.

The best known result ({{As of|2008|lc=on}}) is

:<math>v(n,d)\leq2^n, \, </math>

due to Pfister in 1967.<ref name=Pf1967/>

In complex analysis the Hermitian analogue, requiring the squares to be squared norms of holomorphic mappings, is somewhat more complicated, but true for positive polynomials by a result of Quillen.<ref>{{cite journal | zbl=0198.35205 | last1=Quillen | first1=Daniel G. | title=On the representation of hermitian forms as sums of squares | journal=Invent. Math. | volume=5 | pages=237–242 | year=1968 }}</ref> The result of Pfister on the other hand fails in the Hermitian case, that is there is no bound on the number of squares required, see D'Angelo–Lebl.<ref>{{cite journal | zbl=06028329 | last1=D'Angelo | first1=John P. | last2=Lebl | first2=Jiri | title=Pfister's theorem fails in the Hermitian case | journal=Proc. Am. Math. Soc. | volume=140 | number=4 | pages=1151–1157 | year=2012 | arxiv=1010.3215 }}</ref>

==See also==
* [[Polynomial SOS]]

==References==
{{reflist}}
* {{cite book | editor=Felix E. Browder | editor-link=Felix Browder | title=Mathematical Developments Arising from Hilbert Problems | series=[[Proceedings of Symposia in Pure Mathematics]] | volume=XXVIII.2 | year=1976 | publisher=[[American Mathematical Society]] | isbn=0-8218-1428-1 | first=Albrecht | last=Pfister | authorlink=Albrecht Pfister (mathematician) | chapter=Hilbert's seventeenth problem and related problems on definite forms | pages=483–489 }}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 }}
* {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=[[Springer-Verlag]] | isbn=978-0-387-72487-4 | pages=15–27 | zbl=1130.12001 }}
* {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}

{{Hilbert's problems}}

{{DEFAULTSORT:Hilbert's Seventeenth Problem}}
[[Category:Real algebraic geometry]]
[[Category:Hilbert's problems|#17]]