Sums of squares over totally real fields are rational sums of squares

SUMS OF SQUARES O VER TOT ALL Y REAL FIELDS ARE RA TIONAL SUMS OF SQUARES CHRISTOPHER J. HILLAR Abstract. Let K be a totally real num ber field with Galois closure L . W e prov e that if f ∈ Q [ x 1 , . . . , x n ] is a sum of m squares in K [ x 1 , . . . , x n ], then f is a sum of 4 m · 2 [ L : Q ]+ 1 “ [ L : Q ] + 1 2 ” squares in Q [ x 1 , . . . , x n ]. Moreov e r, our ar gumen t is const ructiv e and gen- eralizes to the case of commutativ e K -algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semi- definite programing pr oblems. 1. Introduction In recent years, techniques from semidefinite pro gramming hav e pro duced nu- merical algo rithms for finding r e presentations of pos itive semidefinite p olynomia ls as sums of squar es. Thes e algo rithms have many applicatio ns in optimization, co n- trol theory , quadratic pr ogra mming, and matrix ana lysis [18, 19, 21, 2 2, 23]. F o r a noncommutativ e application o f these techniques to a famous trace conjecture, see the pape r s [1, 8, 1 3, 16] whic h con tin ue on the work of [9]. One ma jor drawback with these algo rithms is that their output is , in general, nu merical. F or many applications, how ev er, exa ct p olynomial identities are needed. In this regar d, Sturmfels ha s asked whether a repr esentation with rea l co efficients implies one o ver the rationals. Question 1.1 (Sturmfels) . If f ∈ Q [ x 1 , . . . , x n ] is a su m of squar es in R [ x 1 , . . . , x n ] , then is f also a sum of squar es in Q [ x 1 , . . . , x n ] ? It is well-kno wn tha t a p olynomial is a s um o f real p olyno mial sq uares if and only if it can b e written in the form (1.1) f = v T B v , in which v is a column vector of mono mials and B is a rea l p ositive semidefinite (square) matrix [25]; in this c ase, the matrix B is called a Gr am matrix for f . If B happ ens to hav e rational entries, then f is a sum of squar es in Q [ x 1 , . . . , x n ] (this follows fro m a Cholesky factorizatio n argument or from a matrix gener a lization of Lagra ng e’s four s quare theor em [10]). Th us, in the la nguage of q uadratic for ms, Sturmfels is a sking whether the existence of a p ositive semidefinite Gra m matrix for f ∈ Q [ x 1 , . . . , x n ] ov er the reals implies that one exists o v er the ra tionals. 1991 Mathematics Subje ct Classific ation. 12Y05, 12F10, 11E25, 13B24. Key wor ds and phr ases. rational sum of squares, semidefinite programming, totally real num- ber field. Supported under a National Science F ounda tion Postdoctoral Researc h F ell owship. 1 2 CHRISTOPHER J. HILLAR Although a co mplete answer to Question 1.1 is not k nown, Parrilo and Peyrl ha ve written a n implementation of SOSTOOLS in the algebra pac k ag e Ma caulay 2 tha t attempts to find rational repr esentations of p oly nomials that a re sums o f squares [20]. Their idea is to appr oximate a real Gra m matrix B with rational num b er s and then pro ject bac k to the linear space of so lutions gov erned b y equation (1.1). The following result says that Question 1.1 has a p ositive a nswer in some “generic” sense; it also e x plains the difficult y of finding counterexamples. Theorem 1.2. L et f ∈ Q [ x 1 , . . . , x n ] . If t her e is an invertible Gr am matrix B for f , then ther e is a Gr am m atrix for f with r ational entries. Pr o of. Let B b e a rea l p os itive s emidefinite matrix and v a vector of monomials such that f = v T B v . Co nsider the set L of real symmetric matrice s S = S T = ( s ij ) of the same size as B for which f = v T S v . This s pa ce corres po nds to the solutions of a set of linear equations in the s ij ov er Q . F ro m elemen ta ry linea r algebra (Gaussian elimination), it follo ws that there is an in tege r k such that L = { S 0 + t 1 S 1 + · · · + t k S k : t 1 , . . . , t k ∈ R } for some r ational symmetric matrices S 1 , . . . , S k . The subset of matrices in L tha t are p os itive definite is determined by a finite set of s trict p oly nomial inequalities in the t 1 , . . . , t k pro duced b y setting all the leading principal minors to b e po sitive [11, p. 404]. B y con tinuit y , a real p os itive definite solution B g ua rantees a r ational one, and this c ompletes the pro o f.  R emark 1.3 . The arg ument ab ove s hows that w e ma y find a r a tional Gr a m matrix of the same size as the original Gram matrix B . Is this true ev en if B is not inv ertible? W e suspect not. Although the general case seems difficult, Question 1.1 has a positive a nswer for univ ariate po ly nomials due to results of Landau [15], Pourc he t [24], and (algor ith- mically) Schw eighofer [29]. In fact, Pourc het ha s s hown that a t most 5 p olynomia l squares in Q [ x ] a r e nee de d to r e present every po sitive semidefinite p olynomial in Q [ x ], and this is b est p os s ible. It follows fr om Artin’s solution to Hilber t’s 17th problem [26, Theor em 2 .1.12] that if f ∈ Q [ x 1 , . . . , x n ] is a sum of squares of rational functions in R ( x 1 , . . . , x n ), then it is a sum o f squares in Q ( x 1 , . . . , x n ). Moreover, from the work of V oevods k y on the Milnor conjectures, it is known that 2 n +2 such squares s uffice [14, p. 530]. How ever, the transitio n from rational functions to polynomia ls is often a v e r y del- icate one. F o r instance, no t every p o lynomial tha t is a sum o f squar es o f r a tional functions is a sum o f squares of polynomials [14, p. 398 ]. More ge ne r ally , Stur mfels is interested in the alge br aic degree [1 7] of maximizing a linear functional ov er the space of a ll sum o f squa res r epresentations of a given po lynomial that is a sum of squares. In the special case of Question 1.1, a p ositive answer signifies an algebra ic degree of 1 for this optimiza tion problem. General theor y (for insta nce, T ar ski’s T r ansfer P rinciple for real closed fields [26, Theorem 2.1.10]) reduces Question 1.1 to one in volving real alg ebraic n umbers . In this paper , we pre sent a p os itive answer to this question for a sp ecia l class of fields. Recall that a total ly r e al n umb er field is a finite alge braic e x tension of Q a ll of whose complex embeddings lie entirely in R . F o r ins tance, the field Q ( √ d ) is totally real for positive, in tegr al d . Our main theorem is the follo wing . SUMS OF SQUARES OVER TOT ALL Y REAL FIE LDS 3 Theorem 1.4. Le t K b e a t otal ly r e al numb er fi eld with Galois closure L . If f ∈ Q [ x 1 , . . . , x n ] is a sum of m squ ar es in K [ x 1 , . . . , x n ] , then f is a sum of 4 m · 2 [ L : Q ]+1  [ L : Q ] + 1 2  squar es in Q [ x 1 , . . . , x n ] . Our techniques also generaliz e na turally to the fo llowing situation. Let R b e a commutativ e Q -alge br a and let K b e a tota lly real num b er field. Also , set S := R ⊗ Q K , which we naturally iden tify as a r ing extension of R = R ⊗ Q Q . If f is a sum of the form f = m X i =1 p 2 i , p i ∈ R , then we say that f is a sum of squar es over R . It is a difficult problem to determine those f which are sums of square s o ver R . In this setting, Theore m 1.4 generalizes in the follo wing wa y . Theorem 1.5. L et K b e a total ly r e al numb er fi eld with Galois clo s u r e L . If f ∈ R is a sum of m squar es in R ⊗ Q K , then f is a sum of 4 m · 2 [ L : Q ]+1  [ L : Q ]+1 2  squar es over R . R emark 1.6 . One can vie w Theorem 1.5 as a “go ing-down” theorem [4] for certain quadratic forms ov er the rings R and R ⊗ Q K . W e do no t k now how muc h the factor 2 [ L : Q ]+1  [ L : Q ]+1 2  can b e improv e d up on, althoug h we susp ect that for p oly - nomial rings, it can b e impr ov ed substantially . W e r emark that it is known [2] that arbitrar ily larg e n umbers of squar es are necessary to represent any sum of squar es ov er R [ x 1 , . . . , x n ], n > 1, making a fixed b ound (for a given n ) a s in the ratio na l function case impos s ible. Our pro of of Theorem 1.4 is also constructive. Example 1.7 . Conside r the p olynomial f = 3 − 1 2 y − 6 x 3 + 18 y 2 + 3 x 6 + 12 x 3 y − 6 xy 3 + 6 x 2 y 4 . This p olynomial is a su m of squar es over R [ x, y ] . T o se e this, let α, β , γ ∈ R b e the r o ots of the p olynomial u ( x ) = x 3 − 3 x + 1 . Then, a c omputation r eve als that f = ( x 3 + α 2 y + β xy 2 − 1) 2 + ( x 3 + β 2 y + γ xy 2 − 1) 2 + ( x 3 + γ 2 y + αxy 2 − 1) 2 . Using our te chniques, we c an c onstruct fr om t his r epr esentation one over Q :  x 3 + xy 2 + 3 y / 2 − 1  2 +  x 3 + 2 y − 1  2 +  x 3 − xy 2 + 5 y / 2 − 1  2 +  2 y − xy 2  2 + 3 y 2 / 2 + 3 x 2 y 4 . This ex ample wil l b e r evisite d many times in the s e quel t o il lustr ate our pr o of.  W e b elieve that in Theor em 1 .4 the field K may b e replaced by any real a lgebraic extension of the ra tionals (th us giving a p o sitive a nswer to Sturmfels’ question); how ever, our tech niques do no t readily gener alize to this situation. W e shall discuss the obstructions throughout our presen tation. The or ganization of this pap er is as follows. In Section 2, we set up our no- tation a nd state a weaker (but still sufficient) version of our main theorem. Sec- tion 3 des crib es a matrix fa c torization for V ander monde matrices. This co nstruc- tion is applied in the subsequent section to r e duce the pro blem to the case K = 4 CHRISTOPHER J. HILLAR Q ( √ l 1 , . . . , √ l r ) for po s itive in teger s l k . Finally , the pr o of of Theorem 1.4 is com- pleted in Section 5. F or simplicity of exp osition, we shall fo cus on the po ly nomial version of our main theor e m although it is an easy ma tter to translate the techniques to prov e the mo re general Theorem 1.5. W e would like to thank T. Y. Lam, Bruce Reznick, and Bernd Sturmfels for int e resting discussions ab o ut this problem. W e also thank the anonymous referee for several suggestions that improv ed the expo sition of this work. 2. Preliminaries An equiv ale nt definition of a tota lly rea l num b er field K is that it is a field generated by a r o ot of an irr educible po lynomial u ( x ) ∈ Q [ x ], a ll of whos e zero es are real. F or instance, the field K = Q ( α, β , γ ) = Q ( α ) aris ing in Exa mple 1.7 is a totally rea l (Galois) extension of Q in this sense. A splitting field o f u ( x ) (a Galois closure of K ) is also totally rea l, so we lose no gener ality in ass uming that K is a to ta lly r eal Galois extens io n o f Q . W e will there fo re assume from now o n that K = Q ( θ ) is Galois and that θ is a real alg ebraic num b er, all o f whose conjugates are also r e a l. W e set r = [ K : Q ] and let G be the Galois group Gal( K/ Q ). F or the rest of our dis cussion, we will fix K = Q ( θ ) with these parameters. W e b egin by stating a weak er fo r mulation of Theorem 1.4. F or the purp oses of this work, a r ational s u m of squ ar es is a linear combination o f squares with p ositive rational co e fficient s. Theorem 2. 1. L et K b e a total ly r e al nu mb er field that is Galois over Q . Then for any p ∈ K [ x 1 , . . . , x n ] , the p olynomial f = X σ ∈ G ( σ p ) 2 c an b e written as a r ational su m of 2 [ K : Q ]+1  [ K : Q ]+1 2  squar es in Q [ x 1 , . . . , x n ] . R emark 2.2 . Elements of t he form P σ ∈ G ( σ p ) 2 are also sometimes called tr ac e forms for the field extension K [1 4, p. 21 7 ]. It is elemen tary , but imp ortant that this res ult implies Theorem 1.4. Pr o of of The or em 1.4. Let f = P m i =1 p 2 i ∈ Q [ x 1 , . . . , x n ] b e a sum o f squar es with each p i ∈ K [ x 1 , . . . , x n ]. Summing b o th s ide s of this equation ov e r all actions o f G = Gal( K/ Q ), we ha ve f = 1 | G | m X i =1 X σ ∈ G ( σ p i ) 2 . The conclusions of Theorem 1.4 now follow immediately from Theo rem 2 .1 and Lagra ng e’s four s quare theorem (every p ositive ratio na l num b er is the sum o f at most four squares).  R emark 2.3 . This a veraging ar gument can also b e found in the pa p ers [3, 6]. W e will fo cus our remaining efforts, therefore, on proving Theorem 2.1. SUMS OF SQUARES OVER TOT ALL Y REAL FIE LDS 5 3. V andermonde F a ctoriza tions T o prepar e for the pro o f of Theo rem 2 .1, we describ e a useful matrix factor - ization. It is inspired by Ilyusheckin’s recent pro of [12] that the discrimina nt of a symmetric matrix of indetermina tes is a sum of squa res, although it would no t surprise us if the factorization w as known muc h earlier . Let A = A T be an r × r sy mmetric matrix over a field F of characteristic not equal to 2, and let y 1 , . . . , y r be the eigenv alues of A in an alg ebraic closur e of F . Also, let V r be the V andermonde matrix V r =      1 1 · · · 1 y 1 y 2 · · · y r . . . . . . . . . . . . y r − 1 1 y r − 1 2 · · · y r − 1 r      . The matrix B = V r V T r has as its ( i, j )th en try the ( i + j − 2)th Newton pow er sum of the eigen v alues of A : r X k =1 y i + j − 2 k . Since the trace of A m is also the m th Newton pow er sum of the y k , it follows that we may wr ite B = [tr( A i + j − 2 )] r i,j =1 ∈ F r × r . W e next giv e another factorizatio n of B in the form C C T . Let E ij be the r × r matrix with a 1 in the ( i, j ) entry a nd 0’s elsewher e . A basis for r × r symmetric matr ices is then g iven by the following  r +1 2  matrices: { E ii : i = 1 , . . . , r } ∪ { ( E ij + E j i ) / √ 2 : 1 ≤ i < j ≤ r } . F or example, the “generic ” symmetric 2 × 2 matrix (3.1) A =  x 11 x 12 x 12 x 22  , with en tr ies in the field F = Q ( x 11 , x 12 , x 22 ), is represented in this basis as x 11  1 0 0 0  + x 22  0 0 0 1  + √ 2 · x 12  0 1 / √ 2 1 / √ 2 0  . This bas is is useful since the inner pr o duct of tw o sy mmetr ic matrices P a nd Q with resp ect to this ba sis is simply tr( P Q ), as one ca n easily c heck. Express the p ow er s A m in terms of this basis and place the vectors o f the co- efficients as rows of a matrix C . The entries of the r ×  r +1 2  matrix C w ill be in F [ √ 2]. Our cons tr uction prov es the formal identit y (3.2) V r V T r = [tr ( A i + j − 2 )] r i,j =1 = C C T . Example 3.1 . With A given by (3.1), the factorization r e ads:  1 1 y 1 y 2   1 y 1 1 y 2  =  1 1 0 x 11 x 22 √ 2 · x 12    1 x 11 1 x 22 0 √ 2 · x 12   . Alge br aic al ly, this e quation r efle cts the fact that for a 2 × 2 symmetric matrix A , tr( A ) = x 11 + x 22 , tr( A 2 ) = tr( A ) 2 − 2 det ( A ) = x 2 11 + x 2 22 + 2 x 2 12 . 6 CHRISTOPHER J. HILLAR  In the next s e ction, we will use the matrix facto rization (3 .2) to replace a Gram matrix ov er Q ( y 1 , . . . , y r ) with one ov er a muc h smaller field. 4. Symmetric ma trices with prescribed characteristic pol ynomial Let K = Q ( θ ) b e totally real and Galo is, and set σ 1 , . . . , σ r to b e the elements of Gal( K/ Q ). Given p ∈ K [ x 1 , . . . , x n ], w e ma y express it in the form p = r − 1 X i =0 q i θ i , for elemen ts q i ∈ Q [ x 1 , . . . , x n ]. With this para meterization, the sum X σ ∈ G ( σ p ) 2 = r X j =1 r − 1 X i =0 q i ( σ j θ ) i ! 2 app earing in the statemen t of Theorem 2.1 ma y b e written succinctly as (4.1)    q 0 . . . q r − 1    T      1 · · · 1 σ 1 θ · · · σ r θ . . . . . . . . . ( σ 1 θ ) r − 1 · · · ( σ r θ ) r − 1         1 σ 1 θ . . . ( σ 1 θ ) r − 1 . . . . . . . . . . . . 1 σ r θ . . . ( σ r θ ) r − 1       q 0 . . . q r − 1    . Let V r be the V andermonde matrix app earing in equa tio n (4.1). W e w o uld like to construct a factor ization as in (3.2) to replace the elements of K with n um b er s from Q ( √ 2). T o a pply the techniques of Section 3, how ever, we must find an r × r symmetric ma trix A who se eige nv alues ar e σ 1 θ, . . . , σ r θ (the ro ots of the minimal po lynomial for θ over Q ). A necessary condition is that these num b ers are all real, but we would like a conv er se. Unfortunately , a c o nv erse with matrices over Q is impo ssible. F or the in terested rea der, w e include a pro of of this basic fact. Prop ositi on 4.1. T her e is no r ational, symmetric matrix with char acteristic p oly- nomial u ( x ) = x 2 − 3 . Pr o of. W e a rgue b y wa y of cont radiction. Let A =  a b b c  , a, b, c ∈ Q , and suppose that u ( x ) = det( xI − A ) = x 2 − ( a + c ) x + ( ac − b 2 ) . It follows that there ar e rationa l num b ers a and b such that a 2 + b 2 = 3 . Multiplying by a common denominator, one finds that there must b e integer solutions u, v , w to the diophan tine equation (4.2) u 2 + v 2 = 3 w 2 . Recall from elemen tar y num b e r theor y that a num ber n is the sum of tw o integral squares if and only if ev ery prime p ≡ 3 (mo d 4 ) that appea rs in the prime factor- ization of n appear s to an ev en pow er . This contradiction finishes the pro of.  SUMS OF SQUARES OVER TOT ALL Y REAL FIE LDS 7 If we allow A to contain square r o ots o f rational num b ers, how ever, then there is alwa ys such a sy mmetric A . This is the conten t of a result o f Fiedler [5]. W e include his proo f for c o mpleteness. Theorem 4. 2 (Fiedler) . L et u ( x ) ∈ C [ x ] b e monic of de gr e e r and let b 1 , . . . , b r b e distinct c omplex num b ers such that u ( b k ) 6 = 0 for e ach k . Set v ( x ) = Q r k =1 ( x − b k ) and cho ose any c omplex num b ers d 1 , . . . , d r and δ that satisfy δ v ′ ( b k ) d 2 k − u ( b k ) = 0 , k = 1 , . . . , r . L et d = [ d 1 , . . . , d r ] T and B = dia g( b 1 , . . . , b r ) . Then the symmet r ic matrix A = B − δ dd T has char acteristic p olynomial e qual to u ( x ) . Pr o of. Applying the Sherman-Morriso n formula [7, p. 50] for the determinant of a rank 1 p er turbation of a matrix, w e have det( xI − A ) = det( xI − B ) + δ det( xI − B ) d T ( xI − B ) − 1 d = r Y k =1 ( x − b k ) + δ r X k =1 d 2 k r Y i =1 ,i 6 = k ( x − b i ) . (4.3) Since the monic po lynomial det( xI − A ) and u ( x ) a gree for x = b 1 , . . . , b r , it follows that they are equal.  R emark 4.3 . There a re simpler, tridiagonal ma trices whic h can replace the matrix A (see [28]); howev er, square ro ots are still ne c essary to construct them. The following corollar y allows us to form a rea l symmetric ma tr ix with charac- teristic poly no mial equal to the minimal p olyno mial for θ ov e r Q . Corollary 4. 4. If u ( x ) ∈ Q [ x ] is monic of de gr e e r and has r distinct r e al r o ots, then ther e ar e p ositive r ational num b ers l 1 , . . . , l r and a symmetric matrix A with entries in Q ( √ l 1 , . . . , √ l r ) su ch that the eigenvalues of A ar e t he r o ots of u ( x ) . Pr o of. Let b 1 , . . . , b r − 1 be r ational num b ers such that exactly one b i is (strictly) betw een consecutive r o ots o f u ( x ), and let b r be a ra tio nal n um ber either smaller than the leas t r o ot of u ( x ) o r gr eater than the lar gest r o ot o f u ( x ). Also, set δ ∈ {− 1 , 1 } suc h that l k = δ u ( b k ) /v ′ ( b k ) is p ositive for each k . T he coro lla ry now follows from Theorem 4.2 b y setting d k = √ l k for each k .  Example 4.5. Consider the p olynomial u ( x ) = x 3 − 3 x + 1 fr om Example 1.7. Cho osing ( b 1 , b 2 , b 3 ) = (0 , 1 , 2) and δ = 1 , we have d = [ √ 2 / 2 , 1 , √ 6 / 2] T and A =   − 1 / 2 − √ 2 / 2 − √ 3 / 2 − √ 2 / 2 0 − √ 6 / 2 − √ 3 / 2 − √ 6 / 2 1 / 2   . One c an e asily verify that the char acteristic p olynomial of A is u ( x ) .  Combining Corollary 4.4 and the constr uction found in Section 3, we hav e proved the following theorem. 8 CHRISTOPHER J. HILLAR Theorem 4 .6. L et K b e a total ly r e al Galois extension of Q and set r = [ K : Q ] . Then for any p ∈ K [ x 1 , . . . , x n ] , ther e ar e p ositive inte gers l 1 , . . . , l r such t hat X σ ∈ G ( σ p ) 2 = q T C C T q , in whi ch q is a ve ctor of p olynomi als in Q [ x 1 , . . . , x n ] and C is an r ×  r +1 2  matrix with entries in F = Q ( √ l 1 , . . . , √ l r , √ 2) . T o illustrate the computations p erformed in the pro of of Theorem 4.6, we present the following. Example 4.7. We c ontinue with Example 4. 5. L et α ∈ R b e the r o ot of u ( x ) with α ∈ (1 , 2) . Then, setting β = 2 − α − α 2 and γ = α 2 − 2 , we have u ( x ) = ( x − α )( x − β )( x − γ ) . The Galois gr oup of K = Q ( α ) is cyclic and is gener ate d by the element σ ∈ G such that σ ( α ) = β . If we let v = [ x 3 + 2 xy 2 − 1 , − xy 2 , y − x y 2 ] T , then the factorization obtaine d by The or em 4.6 is given by f = v T         1 − 1 / 2 3 / 2 1 0 2 1 1 / 2 5 / 2 0 − 1 2 0 − √ 6 / 2 √ 6 / 2 0 − √ 3 0         T         1 − 1 / 2 3 / 2 1 0 2 1 1 / 2 5 / 2 0 − 1 2 0 − √ 6 / 2 √ 6 / 2 0 − √ 3 0         v . One c an che ck that t his factorization alr e ady pr o duc es the ra t ional su m of squar es r epr esentation we enc ount er e d in Example 1.7.  W e note that when K is a n a rbitrary num b er field, Galo is ov er Q , our approach still pro duces a res ult similar in spirit to Theo rem 4.6. The only difference is that we must allo w negative integers l k in the statemen t. Theorem 4.8. L et K b e a fi nite Galois ext ension of Q and set r = [ K : Q ] . Then for any p ∈ K [ x 1 , . . . , x n ] , ther e ar e inte gers l 1 , . . . , l r such that X σ ∈ G ( σ p ) 2 = q T C C T q , in whi ch q is a ve ctor of p olynomi als in Q [ x 1 , . . . , x n ] and C is an r ×  r +1 2  matrix with entries in F = Q ( √ l 1 , . . . , √ l r , √ 2) . The following corollar y is the closest w e come to ans wering Sturmfels’ q uestion in the g eneral ca se. It follows from applying The o rem 4 .8 in the s ame wa y that Theorem 4.6 will be used b elow to prov e Theo rem 5.1. Corollary 4.9. L et K b e a finite extension of Q . If f ∈ Q [ x 1 , . . . , x n ] is a sum of squar es over K [ x 1 , . . . , x n ] , then it is a differ enc e of two sums of squar es over Q [ x 1 , . . . , x n ] . Example 4 .10. Consider t he de gr e e 2 field extension K = Q ( i √ 2) , which is the splitting field of u ( x ) = x 2 + 2 . One c an ch e ck that setting ( b 1 , b 2 ) = (0 , 1) , δ = − 1 , and d = [ √ 2 , i √ 3] T in The or em 4.2 pr o duc es the symmetric matrix A =  2 i √ 6 i √ 6 − 2  . SUMS OF SQUARES OVER TOT ALL Y REAL FIE LDS 9 It fol lows that the 2 × 2 V andermonde matrix V 2 as in (4.1) satisfies V 2 V T 2 = C C T , in which C =  1 1 0 2 − 2 2 i √ 3  . This c alculation expr esses the p olynomial f = ( x + i √ 2 y ) 2 + ( x − i √ 2 y ) 2 as the differ enc e f = ( x + 2 y ) 2 + ( x − 2 y ) 2 − 12 y 2 .  5. Proof of Theorem 2.1 In this section, we co mplete the pro o f of our main theor em. The r esults so far show that if f is a sum of m squares in K [ x 1 , . . . , x n ] for a totally real field K , Galois ov er Q , then f is a sum of m ·  [ K : Q ]+1 2  squares in L [ x 1 , . . . , x n ], where L = Q ( √ l 1 , . . . , √ l r , √ 2) for some positive integers l k . The pr o of of Theorem 2.1 is th us complete if we ca n sho w the following. Theorem 5.1. L et l 1 , . . . , l r +1 b e p ositive int e gers and set L = Q ( √ l 1 , . . . , p l r +1 ) . If f ∈ Q [ x 1 , . . . , x n ] is a sum of s squar es in L [ x 1 , . . . , x n ] , then f is a r ational sum of at most s · 2 r +1 squar es in Q [ x 1 , . . . , x n ] . Pr o of. Let l be a p o sitive integer and let L = F ( √ l ) b e a quadra tic extension of a field F of characteristic 0 . W e shall prove: If f ∈ F [ x 1 , . . . , x n ] is a r ational sum of s squa res in L [ x 1 , . . . , x n ], then f is a rational sum o f a t mo st 2 s squar es in F [ x 1 , . . . , x n ]. The theo rem then follo w s by r ep eated application of this fact. If L = F , then there is nothing to prov e. Otherwise, let σ ∈ Gal( L/F ) be such that σ ( √ l ) = − √ l , and let f ∈ F [ x 1 , . . . , x n ] b e a sum of s square s in L [ x 1 , . . . , x n ]: f = s X i =1 p 2 i = 1 2 s X i =1  p 2 i + ( σp i ) 2  . It therefore suffices to prov e that for fixed p ∈ L [ x 1 , . . . , x n ], the element p 2 + ( σ p ) 2 is a rational sum of 2 squares. Finally , writing p = a + b √ l for a, b ∈ F [ x 1 , . . . , x n ], we hav e that ( a + b √ l ) 2 + ( a − b √ l ) 2 = 2 a 2 + 2 lb 2 . This completes the pr o of.  References [1] S. Burgdorf , Sums of Hermitian Squar es as an Appr o ach to the BMV Conje ctur e , preprint. [2] M . D . Choi, Z. D. Dai, T. Y. Lam, B. Rez ni c k, The Pythagor as numb er of some affine algebr as and lo c al algebr as . J. Reine A ngew. Math. 336 (1982), 45–82. [3] M . D. Choi, T. Y. Lam, B. Reznic k, Even symmetric sextics , Math. Z., 19 5 (1987), 559–580. 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Dep ar tment of Mathema tics, Texas A&M University, College St a tion, TX 7784 3 E-mail addr ess : chillar@math .tamu.ed u