Determinantal Representations and the Hermite Matrix

DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX TIM NETZER, D ANIEL PLA UMANN, AND ANDREAS THOM Abstra ct. W e consider the problem of writing real p olynomials as determinan ts of sym- metric linear matrix p olynomials. This problem of algebraic geometry , whose ro ots go back to the nineteen th cen tury , has recently receiv ed new attention from the viewpoint of conv ex optimization. W e relate the question to sums of squares decomp ositions of a certain Her- mite matrix. If some pow er of a p olynomial admits a definite determinan tal representation, then its Hermite matrix is a sum of squares. Con v ersely , we show ho w a determinantal represen tation can sometimes b e constructed from a sums-of-squares decomp osition of the Hermite matrix. W e finally show that definite determinantal representations alwa ys exist, if one allo ws for denominators. Intr oduction A p olynomial p ∈ R [ x ] in n v ariables x = ( x 1 , . . . , x n ) with p (0) = 1 is called a r e al-zer o p olynomial if p has only real zeros along every line through the origin. The terms hyp erb olic or r e al stable p olynomial are also common and mean essentially the same, but usually for homogeneous p olynomials. The typical example is a p olynomial given by a definite (linear symmetric) determinantal r epr esentation p = det( I + A 1 x 1 + · · · + A n x n ) , where A 1 , . . . , A n are real symmetric matrices and I is the iden tit y . A represen tation of this form is a certificate for b eing a real-zero p olynomial. In other w ords, the fact that p is a real- zero p olynomial is apparen t from the represen tation. A definite determinantal represen tation also provides a description of the rigid ly c onvex r e gion of p . This is the closed connected comp onen t of the origin in the complemen t of the zero-set of p . It is alwa ys con vex, and giv en a definite determinantal represen tation of p , it coincides with the set of p oints where the matrix p olynomial I + A 1 x 1 + · · · + A n x n is p ositiv e semidefinite. Date : Nov ember 27, 2024. 2000 Mathematics Subje ct Classific ation. Primary 11C20, 11E25, 14P10; Secondary 90C22, 90C25, 52B99. Key wor ds and phr ases. determinan tal represen tations, real-zero p olynomials, sp ectrahedra, Hermite matrix, sums of squares. T ra v el for this pro ject was partially supported by the F orsch ungsinitiativ e R e al Algebr aic Ge ometry and Emer ging Applic ations at the Univ ersit y of Konstanz. Daniel Plaumann gratefully ackno wledges support through a F eo dor Lynen return fello wship from the Alexander v on Hum b oldt F oundation. 1 2 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM In recent years, real-zero p olynomials and their determinan tal represen tations ha v e b een studied mostly with a view tow ards con vex optimization, sp ecifically semidefinite and hy- p erb olic programming. In general, one w ould lik e to answ er the following questions: (1) Under what conditions do es a real-zero p olynomial hav e a definite determinan tal represen tation? (2) If suc h a represen tation exists, what is the minimal matrix dimension and ho w can the represen tation b e computed effectiv ely? (3) If no suc h representation exists, what other certificates for b eing a real-zero p olyno- mial are av ailable? Question (1) is the most immediate and has consequently received the most attention. It ties in with the theory of determinan tal hypersurfaces in complex algebraic geometry , whose ro ots go back to the nineteen th cen tury . Arguably the most imp ortan t mo dern results are the Helton-Vinnik ov theorem in [ 7 ], which giv es a p ositive answer for n = 2 , and Brändén’s negativ e results in higher dimensions in [ 4 ]. Since there are v arious subtle v ariations of the question, it is not alw ays easy to figure out what is known and what is not; w e give a v ery brief o verview after the in tro duction below. Question (2), which should b e of interest for practical purp oses, has not b een studied v ery systematically so far. Ev en in the case n = 2 , the classical approach of Dixon for constructing determinan tal represen tations is quite algorithmic in nature but hard to carry out in practice (see [ 5 ], or [ 14 ] for a more mo dern presen tation.) One approach to Question (3) is to study the determinan tal representabilit y of a suitable p o w er or multiple of p if no representation for p exists. This is motiv ated by the Generalized Lax Conjecture, as describ ed b elo w. On the other hand, the real-zero prop ert y do es not ha v e to b e expressed b y a determinan tal representation. That a p olynomial p in one v ariable has only real ro ots is equiv alen t to its Hermite matrix b eing p ositive semidefinite. This is a symmetric real matrix asso ciated with p , whic h provides one of the classical metho ds for ro ot counting. T o treat the m ultiv ariate case, we use a parametrized v ersion of the Hermite matrix with p olynomial entries. In a typical sums-of-squares-relaxation approach common in p olynomial optimization, w e then ask for the parametrized Hermite matrix H ( p ) to b e a sum of squares, which means that there exists a matrix Q suc h that H ( p ) = Q T Q . (This is called a sum of squares rather than a square, b ecause Q is allo w ed to b e rectangular of any size). This approach has b een used b efore by Henrion in [ 9 ] and b y Parrilo (unpublished) as a relaxation for the real-zero prop ert y , whic h is exact in the t wo-dimensional case. While the Hermite matrix pro vides a practical w a y of certifying the real-zero prop erty , ha ving a definite determinantal representation of p is clearly muc h more desirable, since it DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 3 also yields a description of the rigidly con v ex region by a linear matrix inequalit y . And even if one is only interested in the real-zero prop erty , the multiv ariate Hermite matrix is a fairly un wieldy ob ject compared to the original p olynomial, and a sum-of-squares decomp osition ev en more so. Our main goal is therefore to use a sum-of-squares decomp osition of the parametrized Hermite matrix of a p olynomial p to construct, as explicitly as p ossible, a definite determi- nan tal representation of p , or at least of some m ultiple of p . W e first sho w in Section 1 that a definite determinantal represen tation of some p ow er of p of the correct size alw ays yields a sum-of-squares decomp osition of H ( p ) (Thm. 1.6 ). In Section 2 , we make an attempt at the conv erse. This is partly motiv ated by our experimental finding that the Hermite matrix of the Vámos polynomial, which is the counterexample of Brändén, is not a sum of squares (Example 1.9 ). Note also that in the case n = 2 , where every real-zero p olynomial p ossesses a definite determinan tal represen tation by the Helton-Vinnik o v theorem, the parametrized Hermite matrix can b e reduced to the univ ariate case. It is therefore a sum of squares if and only if it is p ositiv e semidefinite, by a result of Jakub ovič [ 10 ]. Giv en a decomp osition H ( p ) = Q T Q , w e show that a definite determinan tal representation of a m ultiple of p can b e found if a certain extension problem for linear maps on free graded mo dules derived from Q has a solution (Thm. 2.5 ). Giv en Q , the searc h for such a solution amounts only to solving a system of linear equations. This metho d can in principle also b e applied if the sums of squares decomp osition uses denominators. Finally , w e show that by allo wing a sum-of-squares decomp osition with denominators, which exists whenev er H ( p ) is p ositive semidefinite, one can alwa ys obtain a determinantal representation with denominators: Theorem. L et p b e a squar e-fr e e r e al-zer o p olynomial with p (0) = 1 . Ther e exists a sym- metric matrix M whose entries ar e r e al homo gene ous r ational functions of de gr e e 1 such that p = det( I + M ) . The precise statement is given in Thm. 3.1 . A cknow le dgements. W e would like to thank Didier Henrion, Pablo P arrilo, Rainer Sinn, and Cyn thia Vinzant for helpful commen ts and discussions. Kno wn resul ts ◦ F or n = 2 , ev ery real-zero p olynomial of degree d has a real definite determinantal represen tation of matrix size d b y the Helton-Vinnik o v theorem [ 7 ]. ◦ F or n ≥ 3 and d sufficiently large, a simple count of parameters shows that only an exceptional set of p olynomials can hav e a real determinan tal represen tation of 4 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM size d . The question whether every real-zero p olynomial has a definite determinantal represen tation of any size b ecame kno wn as the generalized Lax conjecture. ◦ The generalized Lax conjecture w as dispro v en b y Brändén who ev en sho wed the exis- tence of real-zero p olynomials p suc h that no p ow er p r has a determinantal represen- tation of any size [ 4 ]. His smallest coun terexample, the so-called Vámos p olynomial , is of degree 4 in 8 v ariables (see 1.9 b elo w). ◦ Netzer and Thom [ 11 ] hav e prov ed that only an exceptional set of p olynomials can ha v e a determinan tal representation, even if one allows for matrices of arbitrary size. This is true for n ≥ 3 and d sufficien tly large, or d ≥ 4 and n sufficiently large. They also show that if p is a real-zero p olynomial of degree 2 , then there exists r ≥ 1 such that p r has a determinan tal represen tation. On the other hand, there exists suc h p where one cannot take r = 1 . ◦ Another result of Helton, McCullough and Vinnik o v [ 8 ] (see also Quarez [ 13 ]) sa ys that ev ery real p olynomial has a real symmetric determinantal representation, though not necessarily a definite one. This means that the constant term in the matrix p olynomial cannot be chosen to b e the iden tity matrix in their result. ◦ The most general form of the Lax conjecture sa ys that every rigidly con v ex set is a sp ectrahedron. In terms of determinan tal represen tations, this amoun ts to the follo wing: F or every real-zero p olynomial p there exists another real-zero p olynomial q suc h pq has a real definite determinantal representation and suc h that q is non- negativ e on the rigidly con vex set of p . This conjecture is still wide op en, even without the additional p ositivit y condition on q . Note that if pq has a definite determinan tal represen tation, then q is automatically a real-zero polynomial. 1. The Hermite Ma trix In this section w e in tro duce the parametrized Hermite matrix H ( p ) of a p olynomial. It is p ositive semidefinite at each p oint if and only if p is a real zero p olynomial. If some p o wer of p admits a determinantal represen tation of the correct size, then H ( p ) even turns out to b e a sum of squares of p olynomial matrices. Let p = t d + p 1 t d − 1 + · · · + p d − 1 t + p d ∈ R [ t ] b e a monic univ ariate p olynomial of degree d and let λ 1 , . . . λ d b e the complex zeros of p . Then N k ( p ) = d X i =1 λ k i DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 5 is called the k -th Newton sum of p . The Newton sums are symmetric functions in the ro ots, and can thus b e expressed as polynomials in the co efficien ts p i of p . The Hermite matrix of p is the symmetric d × d matrix H ( p ) := ( N i + j − 2 ( p )) i,j =1 ,...d . It is a Hankel matrix whose en tries are p olynomial expressions in the co efficients of p . Note that H ( p ) = V T V , where V is the V andermonde matrix with co efficients λ 1 , . . . , λ d . The follo wing w ell-known fact go es bac k to Hermite. F or a proof, see for example Theorem 4.59 in Basu, P ollac k and Roy [ 1 ]. Theorem 1.1. L et p ∈ R [ t ] b e a monic p olynomial. The r ank of H ( p ) is e qual to the numb er of distinct zer os of p in C . The signatur e of the Hermite matrix H ( p ) is e qual to the numb er of distinct r e al zer os of p . In p articular, H ( p ) is p ositive definite if and only if al l zer os of p ar e r e al and distinct, and H ( p ) is p ositive semidefinite if an only of al l zer os ar e r e al.  No w let p ∈ R [ x ] b e a p olynomial of degree d in n v ariables x = ( x 1 , . . . , x n ) . The p olynomial p is called a r e al-zer o p olynomial (with resp ect to the origin) if p (0) = 1 and for ev ery a ∈ R n , the univ ariate p olynomial p ( ta ) ∈ R [ t ] has only real zeros. W e wan t to express this condition in terms of a Hermite matrix. W rite p = P d i =0 p i with p i homogeneous of degree i , and let P ( x, t ) = P d i =0 p i t d − i b e the homogenization of p with resp ect to an additional v ariable t . W e consider P as a monic univ ariate polynomial in t and call the Hermite matrix H ( P ) the p ar ametrize d Hermite matrix of p , denoted H ( p ) . Its en tries are p olynomials in the homogeneous parts p i of p . The ( i, j ) -entry is a homogeneous p olynomial in x of degree i + j − 2 . Corollary 1.2. A p olynomial p ∈ R [ x ] with p (0) = 1 is a r e al-zer o p olynomial if and only if the matrix H ( p )( a ) is p ositive semidefinite for al l a ∈ R n . Pr o of. By Theorem 1.1 , H ( p )( a ) is p ositiv e semidefinite for a ∈ R n if and only if the uni- v ariate p olynomial t d p ( a 1 t − 1 , . . . , a n t − 1) has only real zeros. Substituting t − 1 for t , we see that this is equiv alen t to p ( ta ) having only real zeros.  The follo wing is Prop osition 2.1 in Netzer and Thom [ 11 ]. Prop osition 1.3. L et M = x 1 M 1 + · · · + x n M n b e a symmetric line ar matrix p olynomial, and let p = det( I − M ) . Then for e ach a ∈ R n , the nonzer o eigenvalues of M ( a ) ar e in one to one c orr esp ondenc e with the zer os of the univariate p olynomial p ( ta ) , c ounting multiplicities. The c orr esp ondenc e is given by the rule λ 7→ 1 λ .  6 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM Lemma 1.4. L et p ∈ R [ x ] b e a r e al-zer o p olynomial of de gr e e d , and assume that p r = det( I − M ) is a symmetric determinantal r epr esentation of size k , for some r > 0 . Then H ( p ) i,j = 1 r ·  tr  M i + j − 2   , exc ept p ossibly for ( i, j ) = (1 , 1) , wher e H ( p ) 1 , 1 = d and tr  M 0  = k . Pr o of. F or each a ∈ R n , the trace of M ( a ) s is the s -p ow er sum of the nonzero eigenv alues of M ( a ) . These eigenv alues are the inv erses of the zeros of p ( ta ) , by Prop osition 1.3 , but eac h such zero gives rise to r many eigenv alues. Since the zeros of p ( ta ) corresp ond to the in v erses of the zeros of t d p ( t − 1 a ) , the trace of M ( a ) s equals the s -p ow er sum of the zeros of t d p ( t − 1 a ) m ultiplied with r . This pro ves the claim.  Definition 1.5. Let H ∈ Sym d ( R [ x ]) be a symmetric matrix with p olynomial entries. H is a sum of squar es , if there is a d 0 × d -matrix Q with p olynomial entries, such that H = Q T Q . This is equiv alen t to the existence of d 0 man y d -v ectors Q i with p olynomial en tries, suc h that H = P k i =1 Q i Q T i . Theorem 1.6. L et p ∈ R [ x ] b e a r e al-zer o p olynomial of de gr e e d . If a p ower p r admits a definite determinantal r epr esentation of size r · d , for some r > 0 , then the p ar ametrize d Hermite matrix H ( p ) is a sum of squar es. Pr o of. Let p r = det( I − M ) with M of size k = r d , and denote by q ( s ) `m the ( `, m ) -en try of M s . Put Q `m =  q (0) `m , . . . , q ( d − 1) `m  T ∈ R [ x ] d . Then we find k X `,m =1 Q `m Q T `m = k X `,m =1 q ( i − 1) `m q ( j − 1) `m ! i,j =1 ,...,d =  tr( M i − 1 M j − 1 )  i,j =1 ,...,d = r H ( p ) , b y Lemma 1.4 .  R emarks 1.7 . (1) If the determinan tal represen tation of p r is of size k > r d , then H ( p ) b ecomes a sum of squares after increasing the (1 , 1) -en try from d to k /r . This is clear from the ab o ve pro of. (2) It was shown in Netzer and Thom [ 11 ] that if a p olynomial p admits a definite determinan tal representation, then it admits one of size dn , where d is the degree of p and n is the n um b er of v ariables. So if any p o w er p r admits a determinantal represen tation of any size, then H ( p ) is a sum of squares, after increasing the (1 , 1) -en try from d to dn . Note that this is indep endent of r. (3) The determinan t of H ( p ) is the discriminan t of t d p ( t − 1 x ) in t . If H ( p ) = Q T Q , it follo ws from the Cauch y-Binet form ula that the determinant of H ( p ) is a sum of squares in DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 7 R [ x ] . Thus, by the ab o ve theorem, the discriminant of det( tI + M ) in t is a sum of squares, a fact that has long b een kno wn, at least since Borc hardt’s w ork from 1846 [ 3 ]. (4) The sums-of-squares decomp osition of H ( p ) obtained b y Thm. 1.6 from a deter- minan tal represen tation p r = det( I − M ) is extremely sp ecial. In principle, it is p ossible to characterize the decomp ositions of H ( p ) coming from a determinantal representation b y a recurrence relation that they m ust satisfy . But this do es not app ear to b e a promising approac h for finding determinantal representations. Example 1.8 . It was sho wn in Netzer and Thom [ 11 ] that if p is quadratic, a high enough p o w er admits a definite determinantal represen tation of the correct size. Th us H ( p ) is a sum of squares in this case. This can also b e shown directly . W rite p = x T Ax + b T x + 1 with A ∈ Sym n ( R ) and b ∈ R n . Then p is a real-zero p olynomial if and only if bb T − 4 A  0 , as is easily chec k ed. W e find t 2 p ( t − 1 x ) = x T Ax + b T x · t + t 2 , and so we compute H ( p ) = 2 − b T x − b T x x T ( bb T − 2 A ) x ! . W rite bb T − 4 A = P n i =1 v i v T i as a sum of squares of column vectors v i ∈ R n . Set Q =       1 − 1 2 b T x 0 1 2 v T 1 x . . . . . . 0 1 2 v T n x       . Then H ( p ) = 2 · Q T Q .  Example 1.9 . W e consider Brändén’s example from [ 4 ]. It is constructed from the Vámos cub e as sho wn in Figure 1 . Its set of bases B consists of all four elemen t subsets of { 1 , . . . , 8 } that do not lie in one of the five affine h yp erplanes. Define q := X B ∈B Y i ∈ B x i , a degree four p olynomial in R [ x 1 , . . . , x 8 ] . It contains as its terms the pro duct of an y c hoice of four pairwisely different v ariables, except for the following five: x 1 x 4 x 5 x 6 , x 2 x 3 x 5 x 6 , x 2 x 3 x 7 x 8 , x 1 x 4 x 7 x 8 , x 1 x 2 x 3 x 4 . No w p = q ( x 1 + 1 , . . . , x 8 + 1) turns out to b e a real-zero p olynomial, of which Brändén has sho wn that no p ow er has a determinantal representation. 8 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM Figure 1. The Vámos Cub e 1 7 2 5 6 4 3 8 W e can apply the sums-of-squares-test to the Hermite matrix H ( p ) here. Unfortunately , the matrix is too complicated to do the computations b y hand. When using a numerical sums-of-squares-plugin for matlab, suc h as Y almip, the result ho wev er indicates that H ( p ) is not a sum of squares. In view of Theorem 1.6 this shows again that no p ow er of p admits a determinan tal representation. Note that if some p ow er p r has a determinantal representation, then it has one of size 4 r . This w as pro ven by Brändén or follo ws more generally from Netzer and Thom, Theorem 2.7 [ 11 ]. Finally , w e can apply the sums-of-squares-test also to small p erturbations of Brändén’s p olynomial. F or example, p can b e appro ximated as closely as desired b y real-zero p oly- nomials, whic h hav e only simple ro ots on eac h line through the origin (in other words, the Hermite matrix is p ositiv e definite at eac h p oin t a 6 = 0 ). Such a smo othening pro cedure is for example describ e in Nuij [ 12 ]. Still, Y almip rep orts that the Hermite matrix is not a sum of squares, if the approximation is close enough. This is exactly what one exp ects, since the cone of sums of squares of p olynomial matrices is closed, and the Hermite matrix dep ends con tin uously on the p olynomial. 2. A general construction method In this section w e are interested in the conv erse of the ab o v e result. Namely , can a sums- of-squares decomp osition of H ( p ) be used to pro duce a definite determinan tal representation of p or some multiple? W e describ e a metho d to do this, which amounts to only solving a system of linear equations. Let p = 1 + p 1 + · · · + p d ∈ R [ x ] b e a real-zero p olynomial of degree d . Since the matrix H ( p ) is ev erywhere p ositiv e semidefinite, it can be expressed as a sum of squares if one allows denominators in R [ x ] . This generalization of Artin’s solution to Hilb ert’s 17th problem w as DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 9 first prov ed by Gondard and Rib en b oim in [ 6 ]. W e need to mak e a sligh t adjustment to our situation. Lemma 2.1. Ther e exist a matrix p olynomial Q ∈ Mat k × d  R [ x ]  , for some k > 0 , and a homo gene ous non-zer o p olynomial q ∈ R [ x ] such that q 2 H ( p ) = Q T Q . Pr o of. By the original result of Gondard and Rib enboim [ 6 ] there is some non-zero p olyno- mial q ∈ R [ x ] suc h that q 2 H ( p ) = Q T Q for some Q ∈ Mat k × d ( R [ x ]) . W e w an t to make q homogeneous. W rite q = q r + q r +1 + · · · + q R , where eac h q i is homogeneous of degree i , and q r 6 = 0 , q R 6 = 0 . Since the i -th diagonal entry in H ( p ) is homogeneous of degree 2( i − 1) , eac h en try in the i -th column of Q has homogeneous parts of degree b et w een r + i − 1 and R + i − 1 . Let Q min b e the matrix one obtains from Q by c ho osing only the homogeneous part of degree r + i − 1 of eac h entry in each i -th column. Put e Q = Q − Q min and note that all entries in the i -th column of e Q hav e non-zero homogeneous parts only in degrees at least r + i . W e now compute q 2 H ( p ) = Q T min Q min + Q T min e Q + e Q T Q min + e Q T e Q , compare degrees on b oth sides, and find q 2 r H ( p ) = Q T min Q min , as desired.  W e will no w describ e the setup that w e are going to use for the rest of this section. W e fix a represen tation of q 2 H ( p ) = Q T Q as in Lemma 2.1 . As b efore, let P = t d · p ( t − 1 x ) = t d + p 1 t d − 1 + · · · + p d ∈ R [ x, t ] , and consider the free R [ x ] -mo dule A = R [ x, t ] / ( P ) ∼ = d − 1 M i =0 R [ x ] · t i ∼ = R [ x ] d . Since P is homogeneous, the standard grading induces a grading on A . W e shift this grading b y r , the degree of q , and obtain a grading with deg( t i ) = r + i for i = 0 , . . . , d − 1 . This turns A in to a graded R [ x ] -mo dule, where R [ x ] is equipp ed with the standard grading. F urthermore, we equip A with a symmetric R [ x ] -bilinear and R [ x ] -v alued map h· , ·i p defined b y h f , g i p := f T  q 2 H ( p )  g , for f = ( f 1 , . . . , f d ) T and g = ( g 1 , . . . , g d ) T in A . Next, consider the map L t : A → A giv en b y m ultiplication with t . This is an R [ x ] -linear map whic h we can compute with resp ect to our chosen basis: L t : ( f 1 , . . . , f d ) T 7→ ( − p d f d , f 1 − p d − 1 f d , . . . , f d − 1 − p 1 f d ) T . 10 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM Note that L t is of degree 1 with resp ect to the grading, i.e. deg  L t ( f )  = deg ( f ) + 1 . W e iden tify L t with the matrix that represen ts it, so that L t =      0 0 0 − p d 1 0 0 − p d − 1 0 . . . 0 . . . 0 · · · 1 − p 1      , whic h is exactly the c omp anion matrix of P , view ed as a univ ariate p olynomial in t . It is w ell known and easy to see that P is the c haracteristic polynomial of L t , so that det ( I − L t ) = p Lemma 2.2. The line ar map L t is self-adjoint with r esp e ct to h· , ·i p , i.e. hL t f , g i p = h f , L t g i p holds for al l f , g ∈ A. Pr o of. W e may divide by q 2 on b oth sides and hence assume that q = 1 . It is enough to sho w hL t e i , e j i p = h e i , L t e j i p for all i, j , where e i is the i -th unit v ector. F or i, j < d , this follo ws from the fact that H ( p ) is a Hank el matrix. F or i = j = d , it is clear from symmetry . So assume j < i = d . W e find hL t e d , e j i p = − d X i =1 p d − i +1 e i H ( p ) e j = − d X i =1 p d − i +1 N i + j − 2 , where N k is the k -th Newton sum of P . On the other hand, w e compute h e d , L t e j i p = h e d , e j +1 i p = N d + j − 1 . In conclusion, we ha v e to show that d X i =0 p d − i N i + j − 1 = 0 , where w e ha ve set p 0 = 1 . This statement is equiv alent to P d i =0 p i N k − i = 0 , where k = d + j − 1 ≥ d . This last equation, ho wev er, follows immediately from the Newton iden tity k p k + P k − 1 i =0 p i N k − i = 0 , where we let p k = 0 for k > d .  Let B = R [ x ] k . The k × d -matrix Q in the decomp osition of H ( p ) describ es an R [ x ] - linear map A = R [ x ] d → B , f 7→ Q f . F rom the degree structure of H ( p ) , we see that each en try in the i -th column of Q is homogeneous of degree r + i − 1 . So Q is of degree 0 with resp ect to the canonical grading on B . Lemma 2.3. DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 11 (1) If p is squar e-fr e e, then Q : A → B is inje ctive. (2) W e have h f , g i p = hQ f , Q g i for al l f , g ∈ A . In other wor ds, Q is an isometry, taking h , i p to the c anonic al biline ar form h , i on B . Pr o of. (2) is immediate from the fact that q 2 H ( p ) = Q T Q . (1) If Q f = 0 , then 0 = hQ f , Q f i = h f , f i p = q 2 · f T H ( p ) f . F or each a ∈ R n for whic h p ( ta ) has only distinct ro ots, the matrix H ( p )( a ) is p ositive definite. So f ( a ) = 0 for generic a , and thus f = 0 .  Time for a brief summary of what we hav e done so far. Setup 2.4. ◦ Let p ∈ R [ x ] b e a real-zero p olynomial of degree d with p (0) = 1 , and let H ( p ) b e its parametrized Hermite matrix. Fix a decomp osition q 2 H ( p ) = Q T Q , where q is homogeneous of degree r and Q is a matrix of size k × d with entries in R [ x ] . ◦ W e hav e equipp ed the free mo dule A = R [ x ] d with a particular grading and with a bilinear form h , i p : A × A → A . ◦ Let B = R [ x ] k b e equipp ed with the canonical bilinear form and the canonical grad- ing. ◦ The map Q : A → B is an isometry and of degree 0 . ◦ Let L t b e the companion matrix of t d p ( t − 1 x ) with resp ect to t , so that det( I − L t ) = p. The map L t : A → A is self-adjoint with resp ect to h· , ·i p and of degree 1 . The follo wing is our main resul t. Theorem 2.5. L et p ∈ R [ x ] b e a squar e-fr e e r e al-zer o p olynomial of de gr e e d with p (0) = 1 . Assume that ther e exists a homo gene ous symmetric line ar matrix p olynomial M of size k × k such that the fol lowing diagr am c ommutes: R [ x ] d = A Q / / L t   B = R [ x ] k M   R [ x ] d = A Q / / B = R [ x ] k Then p divides det( I − M ) . 12 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM R emark 2.6 . Note that the ab ov e describ ed setup exactly means that w e can hop e for such a linear symmetric M to exist. Indeed the "strange" symmetry of L t is transformed in to the standard symmetry by Q , and the "strange" grading is translated to the standard grading. Pr o of. F or generic a ∈ R n , the map Q ( a ) is injective b y Lemma 2.3 . Therefore, all eigen v alues of L t ( a ) are also eigen v alues of M ( a ) . The eigenv alues of L t ( a ) are precisely the zeros of P ( t, a ) , i.e. the inv erses of the zeros of p ( ta ) . So q = det( I − M ) v anishes on the zero set of p , by Prop osition 1.3 . Since p is a square-free real zero p olynomial, the ideal ( p ) generated b y p in R [ x ] is real-radical (see Bo chnak, Coste and Ro y [ 2 ], Theorem 4.5.1(v)). It follows that q is con tained in ( p ) , in other words p divides q .  R emark 2.7 . Whether there exists such M can b e decided by solving a system of linear equations. Indeed, set M = x 1 M 1 + · · · + x n M n , where the M i are symmetric matrices with indeterminate en tries. The equation MQ = QL t of matrix p olynomials can b e considered en trywise, and comparison of the co efficients in x giv es rise to a system of linear equations in the entries of the M i . Example 2.8 . Let p ∈ R [ x ] be quadratic. W rite p = x T Ax + b T x + 1 with A ∈ Sym n ( R ) and b ∈ R n . W e hav e seen in Example 1.8 that H ( p ) admits a sums of squares decomposition if p is a real-zero p olynomial, giv en b y the matrix Q = √ 2 ·      1 − 1 2 b T x 0 1 2 v T 1 x . . . . . . 0 1 2 v T n x      if bb T − 4 A = P n i =1 v i v T i . It is now easy to find a homogeneous linear matrix p olynomial M that mak es the diagram in Theorem 2.5 commute, namely w e can take M = 1 2 ·      − b T x v T 1 x · · · v T n x v T 1 x − b T x 0 0 . . . 0 . . . 0 v T n x 0 0 − b T x      . The resulting determinantal representation is det ( I − M ) =  1 + 1 2 · b T x  n − 1 · p. T o giv e an explicit example, consider p = ( x 1 + √ 2) 2 − x 2 2 − x 2 3 − x 2 4 − x 2 5 , whic h itself do es not admit a determinan tal representation (by Netzer and Thom [ 11 ]). The pro cedure DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 13 just describ ed no w giv es rise to the linear matrix p olynomial M =         − √ 2 x 1 x 1 x 2 x 3 x 4 x 5 x 1 − √ 2 x 1 0 0 0 0 x 2 0 − √ 2 x 1 0 0 0 x 3 0 0 − √ 2 x 1 0 0 x 4 0 0 0 − √ 2 x 1 0 x 5 0 0 0 0 − √ 2 x 1         and finally det( I − M ) = (1 + √ 2 x 1 ) 4 · p. Example 2.9 . There are also examples where no suitable M exists. W e are grateful to Rainer Sinn and Cynthia Vinzan t for helping us find this example. Consider the plane cubic p = ( x 1 − 1) 2 ( x 1 + 1) − x 2 2 . One computes H ( p ) =   3 x 1 3 x 2 1 + 2 x 2 2 x 1 3 x 2 1 + 2 x 2 2 x 3 1 + 3 x 1 x 2 2 3 x 2 1 + 2 x 2 2 x 3 1 + 3 x 1 x 2 2 3 x 4 1 + 8 x 2 1 x 2 2 + 2 x 4 2   = Q T Q , where Q =     0 x 2 ax 1 x 2 0 − x 2 bx 1 x 2 √ 2 √ 2 x 1 √ 2( x 2 1 + x 2 2 ) 1 − x 1 x 2 1     and a = 1 2 ( √ 7 + 1) , b = 1 2 ( √ 7 − 1) . The equation MQ = QL t has 12 entries, eac h of whic h gives rise to sev eral linear equations b y comparing co efficients in x. One can chec k that already the equations obtained from the first tw o ro ws of MQ = QL t are unsolv able. 3. Ra tional represent a tions of degree one There is alwa ys a w a y to make the diagram from the last section commute, if one allows for r ational linear matrix p olynomials. This will lead to rational determinantal represen ta- tions, as describ ed no w. Let p b e a square-free real-zero p olynomial. Since the parametrized Hermite matrix H ( p ) ev aluated at a p oin t a ∈ R n is p ositiv e definite for generic a , the matrix p olynomial H ( p ) is in vertible ov er the function field R ( x ) . Recall that the degree of a rational function f /g ∈ R ( x ) is defined as deg( f ) − deg( g ) . F urthermore, we say that f /g is homogeneous if b oth f and g are homogeneous, not necessarily of the same degree. Equiv alently , f /g is homogeneous of degree d if and only if ( f /g )( λa ) = λ d ( f /g )( a ) holds for all a ∈ R n with g ( a ) 6 = 0 . 14 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM Theorem 3.1. L et p b e a squar e-fr e e r e al-zer o p olynomial. W rite q 2 H ( p ) = Q T Q with q homo gene ous as in L emma 2.1 and let M := q − 2 QL t H ( p ) − 1 Q T . The matrix M is symmetric with entries in R ( x ) homo gene ous of de gr e e 1 , and satisfies det( I − M ) = p. Pr o of. Abbreviate H ( p ) b y H and L t b y L . By Sylvesters determinan t theorem, w e ha v e det( I k − AB ) = det( I d − B A ) for any matrix p olynomials A of size k × d and B of size d × k . In our situation, this yields det( I k − M ) = det( I k − q − 2 QLH − 1 Q T ) = det( I d − q − 2 LH − 1 Q T Q ) = det( I d − L ) = p. W e find M T = q − 2 Q ( H − 1 ) T L T Q T = q − 2 QLH − 1 Q T = M , where w e hav e used L T H = H T L , whic h is Lemma 2.2 . Thus M is symmetric. Let r b e the degree of q . By examining the degree structure of q 2 H , w e find Q ( λa ) = Q ( a ) · diag( λ r , λ r +1 , . . . , λ r + d − 1 ) H ( λa ) = diag ( λ 0 , . . . , λ d − 1 ) · H ( a ) · diag ( λ 0 , . . . , λ d − 1 ) L ( λa ) = diag( λ d , . . . , λ 1 ) · L ( a ) · diag( λ − d +1 , λ − d +2 . . . , λ 0 ) for all a ∈ R n and λ 6 = 0 . Hence for all a ∈ R n for which H ( a ) is inv ertible and q ( a ) 6 = 0 , and all λ 6 = 0 , we hav e M ( λa ) = λ − 2 r q ( a ) − 2 Q ( a ) · diag( λ r , . . . , λ r + d − 1 ) · diag( λ d , . . . , λ ) L ( a ) · diag( λ − d +1 , . . . , λ 0 ) · diag ( λ 0 , . . . , λ − d +1 ) · H − 1 ( a ) · diag( λ 0 , . . . , λ − d +1 ) · diag( λ r , . . . , λ r + d − 1 ) · Q T ( a ) = λ − 2 r q ( a ) − 2 Q ( a ) λ r + d L ( a ) λ − d +1 H − 1 ( a ) λ r Q T ( a ) = λ · M ( a ) .  R emark 3.2 . Note that a representation p = det ( I − M ) as in Theorem 3.1 gives an alge- braic certificate for p b eing a real-zero p olynomial. Since p ( ta ) = det ( I − t M ( a )) , using homogeneit y , the zeros of p ( ta ) are just the in verses of the eigen v alues of M ( a ) . Since M is symmetric, all of these zeros are real. Theorem 3.1 now states that such an algebraic certificate exists for e ach real-zero polynomial p . DETERMINANT AL REPRESENT A TIONS AND THE HERMITE MA TRIX 15 Example 3.3 . Consider the quadratic polynomial p = ( x 1 + 1) 2 − x 2 2 − x 2 3 − x 2 4 . W e ha ve H =  2 − 2 x 1 − 2 x 1 2( x 2 1 + x 2 2 + x 2 3 + x 2 4 )  = Q T Q with Q T =  √ 2 0 0 0 − √ 2 x 1 √ 2 x 2 √ 2 x 3 √ 2 x 4  , whic h results in M =      − x 1 x 2 x 3 x 4 x 2 − x 1 x 2 2 x 2 2 + x 2 3 + x 2 4 − x 1 x 2 x 3 x 2 2 + x 2 3 + x 2 4 − x 1 x 2 x 4 x 2 2 + x 2 3 + x 2 4 x 3 − x 1 x 2 x 3 x 2 2 + x 2 3 + x 2 4 − x 1 x 2 3 x 2 2 + x 2 3 + x 2 4 − x 1 x 3 x 4 x 2 2 + x 2 3 + x 2 4 x 4 − x 1 x 2 x 4 x 2 2 + x 2 3 + x 2 4 − x 1 x 3 x 4 x 2 2 + x 2 3 + x 2 4 − x 1 x 2 4 x 2 2 + x 2 3 + x 2 4      . References [1] S. 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Vinnik ov, Complete description of determinantal r epr esentations of smo oth irr e ducible curves , Linear Algebra Appl. 125 (1989), 103–140. ↑ 2 16 TIM NETZER, D ANIEL PLAUMANN, AND ANDREAS THOM Tim Netzer, Universit ä t Leipzig, Germany E-mail addr ess : netzer@math.uni-leipzig.de D aniel Plaumann, Universit ä t K onst anz, Germany E-mail addr ess : Daniel.Plaumann@uni-konstanz.de Andreas Thom, Universit ä t Leipzig, Germany E-mail addr ess : thom@math.uni-leipzig.de