资源描述：
《abstract impossibility of c ∞ variation or formal power series variation in solutions to h》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。
1、ImpossibilityofC∞variationorformalpowerseriesvariationinsolutionstoHilbert’s17thproblem⋆CharlesN.Delzell1DepartmentofMathematics,LouisianaStateUniversity,BatonRouge,Louisiana70803,USAAbstractNomatterhowapositivesemidefinitepolynomialf∈R[X1,...,Xn]isrepresented(accordingtoE.Artin’s19
2、26solutiontoHilbert’s17thproblem)intheformf=Ppiri2(with0≤pi∈Randri∈R(X1,...,Xn)),thepiandthecoefficientsofthericannotbechosentodependinaC∞(i.e.,infinitelydifferentiable)manneruponthecoefficientsoff(unlessdegf≤2);formalpowersseriesvariationisalsoimpossible.Thisanswersaquestionwehadraisedi
3、n1990(Contemp.Math.,Vol.155,AMS,1994,pp.107–17),wherewehadalreadyshownthatrealanalyticvariationwasimpossible;andGonzalez-VegaandLombardi(Math.Z.225(3)(1997),427–51)thenshowedthatforeveryfixed,finiter∈N,Crvariationispossible,improvingupontheirandtheauthor’sresultthatcontinuous,piecewi
4、se-polynomialvariationispossible.Keywords:C∞functions,sumsofsquares,basicclosedsemianalyticsets,positivesemidefinitepolynomials,Hilbert’s17thproblem,formalpowerseries,Weierstraßpolynomials1991MSC:14P15(primary),26E10,11E10,12D15,26C15(secondary)1.IntroductionSupposen∈N:={0,1,...},X:
5、=(X1,...,Xn)areindeterminates,andf∈R[X]ispsd(positivesemidefinite),i.e.,∀x:=(x,...,x)∈Rn,f(x)≥0.1n⋆Seehttp://at.yorku.ca/cgi-bin/amca/cacv-60foranabstractofthispaper(datedJune1999).Emailaddress:delzell@math.lsu.edu(CharlesN.Delzell).1PartiallysupportedbyNSF-DMS-ANT-9401509.Preprints
6、ubmittedJan.17,2003Hilbert’s17thproblem[Hi1900]wastoprovethatwecanalwayswritesuchanfintheformX2f=ri,iforsomeri∈R(X).E.Artinsolvedthisproblemin[A1926],andwentontoprovethatifK⊆Risasubfield,f∈K[X],andfispsd,thenwecanwriteX2f=piri,(1.0.1)forsomeri∈K(X)andpi∈Ksuchthatpi≥0.³Parametrizatio
7、nofHilbert’s17thproblem:´Nowletd∈N,letm:=mnd=n+dn,letC:=(C1,...,Cm)beindeterminates,andletfnd:=fnd(C;X)∈Z[C;X]bethegeneralpolynomialofdegreedinXwithcoefficientsC:Xαfnd=Cj(α)X,
8、α
9、≤dnPαα1αnwhereα=(α1,...,αn)∈N,
10、α
11、=αi,X=X1···Xn,andjisanyfixedbijection:j:=jnd:{α
12、
13、α
14、≤d}→{1,...,m}.Writingc=
15、(c,...,c)∈Rm,let1mP={c∈Rm
16、
