Minkowski s inequality and sums of squares

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1 Cet Eur J Math DOI: 10478/s Cetral Euroea Joural of Mathematcs Mows s equalty ad sums of squares Research Artcle Péter E Freel 1, Péter Horváth 1 1 Deartmet of Algebra ad Number Theory, Mathematcs Isttute, Faculty of Scece, Eötvös Lorád Uversty, Pázmáy Péter sétáy 1/C, Budaest 1117, Hugary Receved 5 February 013; acceted 1 Jue 013 Abstract: Postve olyomals arsg from Murhead s equalty, from classcal ower mea ad elemetary symmetrc mea equaltes ad from Mows s equalty ca be rewrtte as sums of squares MSC: 6D05 Keywords: Algebrac equaltes Mows s equalty Postve olyomals Sums of squares Versta S z oo 1 Itroducto May of the most mortat equaltes mathematcs are, or ca be reformulated as, algebrac equaltes A algebrac equalty s oe that asserts that some gve olyomal s oegatve everywhere or oegatve o some secfed set A olyomal f R[x 1,, x ] that s oegatve everywhere s ot ecessarly a sum of squares of olyomals Ths fact was cojectured by Mows ad roved by Hlbert The smlest ow examle has bee gve by Motz However, f the equalty f 0 s classcal ad famous eough, the f usually turs out to be reresetable as a sum of squares, although such a reresetato s ot always easy to fd For examle, the most stadard roof of the Cauchy Schwarz equalty s ot the oe that rewrtes the dfferece of the two sdes as a sum of squares, but such a rewrtg s ossble ad almost as well ow More terestgly, the equalty betwee the arthmetc ad the geometrc meas also has such a roof, as was demostrated by Hurwtz [5] 1891 The aer of Fujsawa [3] gves umerous further examles of ths heomeo Such a urely algebrac roof of a algebrac equalty, eve f t s ot the smlest roof, gves some extra uderstadg of why the equalty must be true E-mal: freel@cseltehu E-mal: horvatheter17@gmalcom 510 Uauthetcated Dowload Date 5/3/18 5:43 PM

2 PE Freel, P Horváth I the reset ote, we gve square sum decomostos of ostve olyomals arsg from the equaltes lsted below I each of these, the varables x, X, Y are meat to be oegatve reals The equalty 1 q x q betwee ower meas holds for ay real exoets q > 0, ad ca be rewrtte as a algebrac equalty whe ad q are tegers Lyauov s more geeral equalty r x q x 1 x q r holds for ay real exoets q r 0, ad s a algebrac equalty whe, q ad r are tegers Note that the secal case r 0 s the recedg ower mea equalty Maclaur s equalty 1 q q 1 1 << q x 1 x q x r q << x 1 x betwee elemetary symmetrc meas holds for tegers q 1, ad ca be rewrtte as a algebrac equalty Note that the secal case q 1, s the equalty betwee the arthmetc ad the geometrc meas A Lyauov tye geeralzato of Maclaur s equalty r r x 1 x q q 1 << q r q x 1 x 1 << q r q q x 1 x r r 1 << r for the elemetary symmetrc meas holds for tegers q r 0, ad s a algebrac equalty Note that the secal case r 0 s Maclaur s equalty, ad the secal case r q 1, q + 1 s Newto s well-ow equalty Ths latter secal case s easly see to mly the geeral case Mows s equalty sueraddtvty of the geometrc mea: X + Y X + Y 1 The treatmet of all of these wll rely o rewrtg Murhead s equalty as a sum of squares see Lemma 31 Ths equalty s more techcal ad so we ostoe t to Secto 3 Noegatve varables Classcal equaltes ofte volve oegatve real varables as oosed to real varables Note that all of our examles above have bee stated for oegatve varables, although some secal cases the oegatvty assumto ca be droed I the settg of oegatve varables, the sutable aalog of the semrg of sums of squares s the semrg S S ε 1 0 ε 0 r ε j : r ε s a sum of squares R[x 1,, x ] x ε j Uauthetcated Dowload Date 5/3/18 5:43 PM 511

3 Mows s equalty ad sums of squares It s mmedately see that S s deed a semrg, e, t s closed uder addto ad multlcato I fact, S s the semrg geerated by the varables x 1,, x ad by the squares of all olyomals Note that S mles that s oegatve for x 1,, x 0, but ot coversely Clearly, s oegatve for x 1,, x 0 f ad oly f x 1,, x s oegatve everywhere The relevace of the semrg S s exlaed by the followg lemma Lemma 1 Let R[x 1,, x ] The S f ad oly f x 1,, x s a sum of squares R[x 1,, x ] To arecate the results roved later sectos of ths aer, the trval oly f statemet wll be mortat We cluded the more dffcult f statemet to mae the cture comlete Proof The oly f art s trval For the f art, we cosder the lear oerator R: R[x 1,, x ] R[x 1,, x ] that mas the moomal x j j to tself f all j are eve ad mas t to zero otherwse We assume that x1,, x s a sum of squares, e, there exst olyomals r such that qx 1,, x x 1,, x r x 1,, x I r we grou terms accordg to the arty of the exoets of x 1,, x We defe the olyomals r,ε for each ε ε 1,, ε {0, 1} so that r x 1,, x ε 1 0 r,ε x1,, x ε 0 x ε j j Aly R to r, the Rr x 1,, x ε 1 0 r,εx 1,, x ε 0 x ε j j Hece, x1,, x qx 1,, x Rqx 1,, x Rr x 1,, x ε 1 0 r,εx 1,, x ε 0 x ε j j Therefore, whece S x 1,, x ε 1 0 r,εx 1,, x ε 0 x ε j j, 51 Uauthetcated Dowload Date 5/3/18 5:43 PM

4 PE Freel, P Horváth 3 Meas We ow wsh to geeralze a few results of Fujsawa [3] cocerg ower mea ad elemetary symmetrc mea equaltes Fx a oegatve teger d, the degree of the homogeeous olyomals we wll be loog at Let us cosder the set of arttos of d to at most arts such a artto s a wealy decreasg -term sequece of oegatve tegers addg u to d There s a stadard artal order o ths set Frst of all, we wrte α β f, for some dces < l, we have β α 1, β l α l + 1 ad β α for, l The, we defe the artal order to be the reflexve trastve closure of, e, α β f ad oly f there exsts N 0 ad a sequece of arttos α α 0 α 1 α N β We wrte α β f α β ad α β We meto, but wll ot mae use of, the well-ow fact that α β holds f ad oly f α α β β for all We ow troduce the Reyolds oerator R of the symmetrc grou S For a olyomal f R[x 1,, x ], let Rfx 1,, x 1 fx σ1,, x σ! σ S We troduce the moomal x α x α j j ad defe the ormalzed moomal symmetrc fucto [α] Rx α We have Murhead s equalty If the arttos α ad β satsfy α β, the [α]x 1,, x [β]x 1,, x for all oegatve x 1,, x The followg lemma wll be crucal for the sequel Lemma 31 Murhead s equalty rewrtte as sums of squares If the arttos α ad β satsfy α β, the [α] [β] S Proof Sce s the reflexve trastve closure of, ad S s closed uder addto, we may assume that α β Let < l be as the defto of The [α] [β] 1 x R α xα l l + x α l xα l x α 1 x α l+1 l x α l+1 The four-term exresso the er aretheses equals x α 1 l,l x α Ths s S, therefore so s the whole exresso x x l x α x α l l + x α 3 x α l+1 l + + x α l+1 x α 3 l + x α l xα l We ow tur to ormalzed ower sums ad ormalzed elemetary symmetrc olyomals P [, 0,, 0] 1 x E [1,, 1, 0,, 0] }{{} 1 x 1 x 1 < < Note that P 0 E 0 1 Uauthetcated Dowload Date 5/3/18 5:43 PM 513

5 Mows s equalty ad sums of squares Lemma 3 For 1, we have P 1 P +1 P P S ad E E E 1 E +1 S For 1 ad for ths was show by Fujsawa [3] Proof We have P 1 P +1 P P [ 1, 0,, 0] [ + 1, 0,, 0] [, 0,, 0] [, 0,, 0] 1 1 [ +, 0,, 0] + 1 [ + 1, 1, 0,, 0] 1 [ + 1, 1, 0,, 0] [,, 0,, 0] 1 [ +, 0,, 0] [,, 0,, 0] Sce,, 0,, 0 + 1, 1, 0,, 0, the frst statemet follows by the revous lemma Exlctly, we get P 1 P +1 P P 1 R 1 x 1 x For the secod statemet, ut β r,,, 1,, 1, 0,, 0 The }{{}}{{}}{{} r + r +r E E 1 r0 j 1 x j 1 x+ j [β r ] r r For all 0 r 1, we have β r β r+1, so [β r+1 ] [β r ] s S by the revous lemma It wll suffce to fd oegatve costats a r such that 1 E E E 1 E +1 a r [βr+1 ] [β r ] r0 It s easy to see wthout calculato that such oegatve costats exst, but we stll do the calculato order to get a exlct formula Put a a 1 0, the the coeffcet of [β r ] wll be a r 1 a r for all r Therefore what we eed to acheve s 1 a r 1 a r r r 1 1 r 1 r r r + 1 Examg the secal cases r 0 ad r 1, we are led to the cojecture 1 1 r r a r 1 r r A easy calculato shows that ths deed gves the correct a r 1 a r The statemet follows; exlctly, E E E 1 E a r R x 1 x x3 x r+x r+3 x r r0 514 Uauthetcated Dowload Date 5/3/18 5:43 PM

6 PE Freel, P Horváth Theorem 33 ower mea equalty ad Maclaur s equalty rewrtte as sums of squares For q, we have P q P q S ad E q E q S Proof We have P q P q P q P q 0 P q 1 P 1 P +1 P P P q q 1 P +1 P q 1, because the latter exresso equals P q 1 1 q P +1 q 1 P +1 P +1 P q 1 P q P P P q 1 P q q P 1 P P q P q P q P q P q 0 We have thus rereseted P q P q as a S-lear combato of the olyomals P 1 P +1 P P, 1 q 1 The frst clam ow follows from that of the revous lemma We omt the roof of the secod statemet because t s essetally the same Theorem 34 Lyauov s equalty ad the geeralzed form of Maclaur s equalty rewrtte as sums of squares For q r, we have P q r Pr q r S ad Eq r E q r Er q S Proof Oly a slght modfcato of the revous roof s eeded We have P q r Pr q P r q 1 P 1 P +1 P P P q r+1 q 1 P +1 P q 1 r, because the latter exresso equals r+1 P q 1 1 r q P +1 q 1 P +1 P +1 P q q P r+1 1 P 1 1 P 1 r r+1 P q P q 1 r P q r The frst clam follows The roof of the secod statemet s essetally the same P q 1 P P q P q P q r Pr q r 4 Mows s equalty We ow come to our ma result, whch cocers Mows s equalty 1 I ths case, we caot get a olyomal equalty by smly rasg both sdes to some ower We also eed to substtute X x ad Y y to get the -varable olyomal equalty x, y 0 Qx 1,, x, y 1,, y x + y x + y 0 Uauthetcated Dowload Date 5/3/18 5:43 PM 515

7 Mows s equalty ad sums of squares Theorem 41 Mows s equalty rewrtte as a sum of squares We have Q S where S S sce we are dealg wth varables Ths aswers a questo rased by Adrés Cacedo o hs teachg blog [1] Proof We exad Q ad grou the arsg moomals accordg to ther total degree the varables x 1,, x Ths gves Q x y j x y Q 0 I j / I 0 I {1,,} I To facltate otato, we th of the dex set {1,, } as Z/Z, the tegers modulo It wll suffce to rove that Q S for all {0,, } We have Q I Z/Z I 1 1 x t0 I+t j / I+t y j For a fxed set I of dces, deote z t x y x I+t j / I+t I Z/Z I { x y j x y j } t0 I+t j / I+t t0 I+t j / I+t y j The z t S Note that S cotas the olyomal fx 1,, x x x by Lemma 31 Therefore, S cotas the olyomal x 1 x [, 0,, 0] [1, 1,, 1] fz 0,, z zt z t, t0 t0 whece Q S Remar 4 Oe of the uamed referees of ths aer called our atteto to the relevace of bomal squares These are squares of the secal form ax α + bx β, cf [, 4] If we relace sum of squares by sum of bomal squares throughout Sectos 4 of ths aer, cludg the defto of S, the all our lemmas ad theorems rema true Note however that sums of bomal squares do ot form a semrg ad the redefed S s ot a semrg Acowledgemets The frst author s research s artally suorted by MTA Réy Ledület Grous ad Grahs Research Grou Refereces [1] Cacedo A, Postve olyomals, htt://cacedoteachgwordresscom/008/11/11/75-ostve-olyomals/ [] Fdalgo C, Kovacec A, Postve semdefte dagoal mus tal forms are sums of squares, Math Z, 011, 693-4, [3] Fujsawa R, Algebrac meas, Proc Im Acad, 1918, 15, [4] Ghasem M, Marshall M, Lower bouds for olyomals usg geometrc rogrammg, SIAM J Otm, 01,, [5] Hurwtz A, Über de Verglech des arthmetsche ud des geometrsche Mttels, I: Mathematsche Were II: Zahletheore, Algebra ud Geometre, Brhäuser, Basel, 193, Uauthetcated Dowload Date 5/3/18 5:43 PM