; TeX output 2015.11.30:1134 6html: html:f荍t 덑 html: html:* ,!"N cmbx12AR OUNDٚ16-DIMENSIONALQUADRA TICFORMSIN%g cmmi12I2 #|{Y cmr83RA&2 cmmi8q[ pK`y cmr10NIKIT*AUUA.KARPENKO `$ - cmcsc10Abstract.\`W*e0Fdetermine0GtheindexesofallorthogonalGrassmanniansofageneric16- $ dimensionalS;quadraticS:formin b> cmmi10I^ ٓR cmr73 0er cmmi7qEU.Thisisappliedtoshowthatthe3-PsternumbGerofS;the$ form`Iis !", cmsy104. Other`Jconsequencesare:Aanewandcharacteristic-freeproGofofarecentresult$ byChernousov{MerkurjevonpropGersubformsinI^ 2፴q(originallyavqailableincharacteristicUQ$ 0)@aswellasanewand@characteristic-freeproGofofanoldresultbyHomann-Tignol$ andIzhbGoldin-Karpenkoon14-dimensionalquadraticformsinI^ 3፴qL(originallyavqailablein$ characteristic_6=2).W*ealsosuggestanextension_ofthemethoGd,basedoninvestigationof$ thegtopGologicalltrationontheGrothendieckgringofamaximalorthogonalGrassmanian,$ whichUUappliestoquadraticformsofdimensionhigherthan16. b XQ cmr12WVeͷwrorkwithnon-degeneratequadraticformsoverarbitraryelds. Recallthata quadraticform,~similartoaPsterform,iscalleda4@ cmti12generffal
5 html:]for{Jgeneral{Kfactsandterminologyrelatedtoquadraticforms,sespSeciallyfordenitionofaZ(quadratic)ZPsterforminarbitrarycrharacteristic.WVewriteIqR=SIq(Fƹ)fortheWittgroupFofclassesEofevren-dimensionalquadraticformsoveraEeldFƹ.RecallthatIq(Fƹ)isasmoSduleorversthesWittringWƹ(F)ofsclassesofnon-degeneratesymmetricbilinearforms.ThereisaltrationbrysubmoSdulesIq =xI2 1RAq *w(!", cmsy10xI2 2RAq:::C#denedasfollorws:bforanydx1,nI2 dRAq F:=uI 2d)K cmsy8 1(Fƹ)Iq(F),vwhereI (F)uW(F)isthefundamenrtalidealandI 2d 1(Fƹ)isitspSorwer. Let'bSeanevren-dimensionalnon-degeneratequadraticformoveraeldFtandletd1bSean inrtegersuchthatthe Wittclass[']2Iq(Fƹ)isinI2 dRAq+(Fƹ).P Then[']canbSewrittenoasasumnofclassesofgenerald-foldPsterforms.8wTheminimalpSossiblenrumberofthesummandsisdenotedPfVd61(')andcalledthed-PsternrumbSerof',cf.8[!html:14 html:,x9c]. GivrenabaseeldkQŹandapSositiveevenintegerm,weareinterestedtodetermine 0rPf Sd {(m)UR:=sup h'Pf#q۟d(Q(');Xwhere.4'.3runsorver.4m-dimensionalquadraticformsdenedorver.4someeldF|kQandsatisfying[']UR2I2 dRAq+(Fƹ). TVrivially,Pfy19(m)%=m=2foranrym.HAlso,itisknown(andrelativelyeasytoshow)that Pfvߟ26(m)]=\(m 2)=2.In the presenrtpapSer,weconcentrateon the3-PsternumbSerPfk3+(m)whicrhisknowntobSenite.8FinitenessofPfVd61(m)fordUR4isanopenquestion. Classical[aresultsfromthetheoryofquadratic[bformsprorvideuswithvXaluesofPfB3F(m)foramupto12:Pf``3 d(m)aisequalto0formUR<8,}@toa1formUR2f8;10g,}Aandato2formUR=12. ff <