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REPRESENTATION OF RATIONAL POSITIVE REAL FUNCTIONS OF SEVERAL VARIABLES BY MEANS OF POSITIVE LONG RESOLVENT
M. F. Bessmertnyi
Abstract. A rational homogeneous (of degree one) positive real matrix-valued function fz(), z d is presented as the Schur complement of a block of the linear pencil А( z ) z11 A ... zdd A with positive semidefinite matrix coefficients Ak . The partial derivative numerators of a rational positive real function are the sums of squares of polynomials.
Mathematics Subject Classification (2010). 15A22; 47А48; 47А56; 94C05.
Key words: positive real function; sum of squares; long-resolvent representation .
1. Introduction
The long-resolvent Theorem, proved by the author in [3, 4] (see also [5] - [7]), states that every rational mm matrix-valued function f()(,,) z f z1 zd can be represented as the Schur complement 1 f()()()()() z A11 z A 12 z A 22 z A 21 z , (1.1) of the block Az22 () of a linear ()()m l m l matrix-valued function (linear pencil)
A11()() z A 12 z A( z ) A0 z 1 A 1 ... zdd A . (1.2) A21()() z A 22 z If, moreover, fz() satisfies the additional condition
(i) f(,,)(,,) z11 zdd f z z , (ii) f()() zT f z ,
(iii) f(,,)(,,) z11 zdd f z z , \{0}), then one can choose matrices Ak , kd0,1, , to be (i) real (resp. (ii) symmetric, (iii) such that
A0 0 ). Any relation (1.1) is called a long-resolvent representation, since for invertible matrices fz() representation (1.1) takes the form
1 1 T f( z ) ( A0 z 1 A 1 ... zdd A ) , (1.3) where (Im ,0) is m() m l matrix, and Im is the matrix unit. There exist many matrix pencil (1.2) representing the same function .
mm A particular role is a class d of mm matrix-valued functions (1.1) with a T homogeneous positive real matrix pencil A( z ) z11 A ... zdd A , AAAk k k 0 , kd1, , , where positive definiteness is understood in the sense of quadratic forms. Functions of class mm d , and only they, are characteristic functions of passive electric 2m-poles (see [4], [8] and
2 review [1]), containing ideal transformers and d types of elements (each element of the k -th type has an impedance zk ). mm Suppose fz() is a function of the class d . From (1.1) it follows that
1 A11()() z A 12 z Im fzIAwAw(), m 21()() 22 1 ()() wAzz () , A21()() z A 22 z A22()() z A 21 z T where wz, d , ():()ww . Then d f() z f () z ( zk z k )() z A k () z . k1
Since Ak 0, we have
(iv). f( z ) f ( z ) 0 for z d ,
dd where {z : Re z1 0, ,Re zd 0} is open right poly-halfplane. Function fz() is a rational. Than from (iv) it follows that
(v). fz() is holomorphic on d . Any matrix-valued function satisfying conditions (i) - (v) is called a positive real. The class mm d of the rational positive real mm matrix-valued functions f(,,) z1 zd was introduced m m m m by the author in [4] (see also [5]). It's clear that dd . The next question is open: Are conditions (i) - (v) sufficient for belonging rational matrix-valued function to the mm class d ?
mm Else: does a matrix-valued function fz() d have a long-resolvent representation with positive semidefinite matrices Ak 0, kd1, , ?
mm mm The case of d 1 is not interesting: the classes 1 and 1 coincide and consist of T functions of the form f() z z1 A with matrix A such that AAA 0. If d 2 , then m m m m mm we also have 22 . If d 3, then the question of coincidence of the classes d mm and d has remained open till now, with the exception functions of degree 2 and some others. To solve this question, it is required to obtain a convenient characterization of matrix-valued functions of the class . The necessary conditions for belonging fz() to the class were obtained by the author in [4]: there exist rational real matrix-valued functions k ()z , d kd1, , holomorphic on such that kk()()zz , \{0} and
d d f()()() z zk k w k z , wz, . (1.4) k1 The sufficiency of condition (1.4) was proved by Kalyuzhnyi-Verbovetzkyi [17, 18]. He also obtained characterizations of the form (1.4) for various operator generalizations of the class , as well as for the operator-valued Schur-Agler functions in the unit polydisk. In [2], for rational matrix-valued Cayley inner Herglotz-Agler functions over the right poly-halfplane was
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obtained long-resolvent representation (1.1) with matrix pencil (1.2), in which the matrix A0 is skew-Hermitian ()AA00 and the other coefficients Ak are positive semidefinite. Thus, the mm class of Cayley inner rational Herglotz-Agler functions is an extension of the class d . The relation (1.4) requires a representation of non-negative polynomials in the form of a sum of squares of rational functions that are holomorphic in an open right poly-halfplane. Using Artin's solution (see [20]) of Hilbert's 17th problem on the representation of a non-negative rational function of several variables as a sum of squares of rational functions is difficult for two reasons: Artin's theorem says nothing on the location of the singularities of the functions in the decomposition, and, moreover, the proof of the existence of the representation is not constructive. It is interesting to note (J.W. Helton [16]) that each non-commutative “positive” polynomial is the sum of the squares of the polynomials. In [19] (1968) T. Koga considered a similar class of positive real functions d variables (without condition (iii) homogeneity of degree 1). For a given positive function of several variables, a method for the synthesis of a passive electric circuit was proposed. The T. Koga synthesis is carried out in two stages.
At the first stage, the positive real function f(,,) z1 zd , which has a degree nk in each ˆ variable zk , is "replaced" by a positive real function f (,,)1 n of degree in each variable separately. The variables 1,, n , n n1 ... nd divided into groups. If the variables of each k -th group are replaced by a variable , then we get the original positive function . The second stage was based on a modification of Darlington's theorem for functions of several variables. In the univariate case, the Darlington method (see [21]; [14], Chapter V) is based on the representation of a non-negative polynomial as a sum of polynomial squares. T. Koga, as a generalization, suggested
Koga's Sum-of-Squares Lemma. Let f f(,,) x1 xd be a polynomial with real coefficients, n 2 quadratic in each variable. If f 0 for real xi ,id1, , , then fh j1 j , where hj are polynomials linear in each variable and n 2d . As pointed out in 1976 by N.K. Bose [10], Koga's proof is wrong. A counterexample is the non-negative polynomial not representable as a sum of polynomial squares constructed by M.-D. Choi [13]:
222222 222222 Fxy( , ) xy11 xy 22 xy 33 2( xyxy 1122 xyxy 2233 xyxy 1133 ) 2( xy 12 xy 23 xy 31 ) . (1.5) It should be noted that polynomial (1.5) cannot appear when synthesizing a positive real function by the method of T. Koga. In Koga's method the nonnegative polynomial is a partial Wronskian p()() x q x FxWqxpx() [(),()]() qx px () 0 (1.6) kxk xxkk of a pair of polynomials satisfying the condition q( z ), p ( z ) 0 if z d . The polynomial (1.5) cannot be represented in the form (1.6). The representation (1.6) strongly narrows the class of polynomials under consideration. In some cases, behavior of the rational function p()/() z q z allows one to analyze the possibility of representing the Wronskian as a sum of polynomial squares. In particular (see Proposition 8.1), if
4 at least one of the non-negative Wronskians W[ q ( z ), p ( z )] cannot be represented as a sum of zk polynomial squares, then the function p()/() z q z cannot be holomorphic on d . Hence follows a statement that "rehabilitates" the synthesis of a positive real function according to the method of T. Koga.
mm Sum of Squares Theorem. If P()/() z q z d , then the partial Wronskians P()() z q z W[,]()() q P q z P z , kd1, , (1.7) zk zzkk are sums of the squares of polynomials.
mm This theorem made it possible to prove the main result of this article: the class d of mm rational positive real matrix-valued functions coincides with the class d of matrix-valued functions that have a long-resolvent representation with positive semidefinite matrices T AAAk k k 0 , kd1, , . This paper is organized as follows. In Section 2, we explains the terminology and provide preliminary information used in the article. For the sake of completeness of presentation, in Section 3 we recall the simplest properties of functions of the class and properties of the degree reduction operator of a rational function. A criterion for the positivity of a real multiaffine function is obtained in Theorem 3.7. In Section 4, we study Artin's denominator properties of nonnegative forms that cannot be represented as a sum of polynomial squares (PSD not SOS forms). It is proved (Theorem 4.3) that each PSD not SOS form has a minimal Artin's denominator sz() with irreducible factors do not change sign on d . Theorem 5.1 (Product Polarization Theorem) is a certain "modification" of the long-resolvent Theorem. In particular, Theorem 5.1 provides a convenient representation of the partial Wronskians. In Section 6, we study the set of the Gram matrices of a given 2n -form F . The Representation Defect Lemma is proved (Lemma 6.8). A representation of product with one positive semidefinite matrix of the pencil is obtained in Theorem 7.1. The proof is based on Product Polarization Theorem, Representation Defect Lemma and Artin's Theorem. The sum of squares Theorem is proved in Section 8 (Theorem 8.2). As a preliminary, in Proposition 8.1 we study the singularity points of a function with a nonnegative Wronskian that cannot be represented as a sum of squares of polynomials. A long-resolvent representation of a rational positive real function with a positive semidefinite matrix pencil is obtained in Theorem 9.1. This proves that the classes of functions and coincide.
2. Terminology, Designations and Preliminary Information
Let []z be a ring of polynomials in the variables zz1,,d with real coefficients.
We say that p()[] z z is affine in z if degpz ( ) 1, and we say that pz() is multiaffine if k zk it is affine in for all kd1, , .
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Recall that a circular region is a closed or open proper subset of the complex plane, which is bounded by circles (straight lines). In particular, the half-plane is a circular region. We need the following statement about symmetric multiaffine polynomials. 2.1. Grace-Walsh-Szego Theorem (see [12], Theorem 2.12). Let p be a symmetric multiaffine polynomial in n complex variables, let C be an open or closed circular region in , and let zz1,, n be points in the region . Suppose, further, that either deg pn or is convex (or both). Then there exists at least one point C such that p(,,)(,,) z1 zn p .
Any polynomial p()[] z z is called a form ( n -form, a homogeneous polynomial of degree n ) if p(,,)(,,) z11 znn p z z , . Everywhere below, with the exception of specially stipulated cases, homogeneous polynomials are considered.
Let (,,) 1 d be a multi-index, where j are non-negative integers. Then
1 d z z1 zd is the monomial of degree 1 ... d . m Let pzjk () be polynomials. Any matrix P( z ) { pjk ( z )} j,1 k is called a matrix polynomial.
M j The polynomial P() z j1 Bj z , where the coefficients B j are constant matrices, is called a
j M matrix-valued form if {}z j1 is a set of monomials of degree n in variables zz1,, d . Polynomials and qz() are said to be coprime if for at least one of the polynomials pzjk () the polynomials , are coprime. m Let f( z ) { fjk ( z )} j,1 k be a rational matrix-valued function. If is the common denominator of the functions fzjk (), j, k 1, , m , then fjk()()/() z p jk z q z , where pzjk () are polynomials. The rational matrix-valued function will be written in the form f()()/() z P z q z , where is a matrix polynomial and is a scalar polynomial. In fact, division P()/() z q z is the standard operation of multiplying of the matrix Pz() by the number qz()1 .
mm By definition, put {fjk ( z )} j, k 1 / z k { f jk ( z ) / z k } j , k 1 . In the univariate case, coprime polynomials pz(), have no common zeros. The situation is different in the case of several variables (the simplest example zz12/ ).
d Further, Z( p ) z : p ( z ) 0 means the zero set of the polynomial p()[,,] z z1 zd . 2.2. Theorem (see [23], Theorem 1.3.2). Suppose d 1 and , are coprime polynomials in d complex variables such that pq(0) (0) 0. If is a neighborhood of zero in d , then: (a) neither of the sets Zp() and Zq() is a subset of the other; (b) for any a there exists z such that qz( ) 0, p()/() z q z a . In what follows, such a point will be called an unremovable singularity of the function f p/ q . A rational function fz() will be called a regular at a point z0 if it is defined and takes a finite value at the point .
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Any matrix A is called real, if AA , where the bar denotes the replacement of each element of by a complex conjugate number. The symbol AT denotes the transposition operation. If A is a matrix with complex elements, then AA T is the Hermitian conjugate matrix. Any real symmetric mm matrix A is called a positive semidefinite (A 0) if the inequality T m T A 0 holds for all , and a positive definite (A 0) if A 0 for all 0. A matrix-valued form Fz() will be called a PSD form (positive semidefinite) if Fx( ) 0 for x d . A matrix-valued PSD form will be called a SOS form (sum of squares) if F()()() z H z H z T , where the elements of the matrix Hz() (possibly rectangular) are scalar forms.
j M Let {}z j1 be the set of all monomials of degree n in variables zz1,, d . Each 2n -form can be represented as a a z1 11 1M 1 M F()(,,) z z z . (2.1) aa1M MM z M M The matrix Aa{}jk j,1 k is called a Gram matrix of the -form . Gram's matrix is not uniquely determined by the 2n -form. It is known ([22, Theorem 1]) that PSD form Fz() is a SOS form if and only if has a positive semidefinite Gram matrix.
If k is a field, then k(,,) x1 xd denotes the set of rational functions in variables xx1,, d with coefficients from the field . 2.3. Artin's Theorem (see [20, Ch. XI, Corollary 2]). Let be a real field admitting only one ordering, and this ordering is Archimedean. Let, further, f()(,,) x k x1 xd be a rational d function that does not take negative values: fa( ) 0 for all a(,,) a1 ad k , in which fa() is defined. Then fx() there is the sum of squares in .□
Artin's theorem gives a solution to Hilbert's 17th problem. In particular, if F()[,,] z z1 zd is PSD form, then there exists a form sz() such that s()() z2 F z is SOS form. The form will be called Artin's denominator of PSD form Fz().
If is of the SOS form, then is also of the SOS form for each form . The question arises: if is not representable as a sum of squares of forms, then for which forms of will the form also not be SOS form? 2.4. Proposition (see [15, Lemma 2.1]). Let Fx() be a PSD not SOS form and sx() an 2 irreducible indefinite form of degree r in [,,]xx1 d . Then sF is also a PSD not SOS form.
22 Proof. Clearly is PSD. If s F k hk , then for every real tuple a with sa( ) 0 , it 2 2 2 follows that s F( a ) 0. This implies hak ( ) 0 k (since hak () is PSD), and so on the real variety s 0 , we have hk 0 as well. So (using [9, Theorem 4.5.1]), for each k , there exists gk 2 ■ so that hkk sg .This gives Fg k k , which is a contradiction.
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2.5. Corollary. Let Fx() be a matrix-valued PSD not SOS form and sx() an irreducible 2 indefinite form in [,,]xx1 d . Then sF is also a PSD not SOS form. Proof. It suffices to apply the proof of Proposition 2.4 to the diagonal elements of the matrix- valued form s2 F H()() x H x T .■
dd Let {z : Re z1 0, ,Re zd 0} be an open right poly-halfplane. Any polynomial p()[,,] z z1 zd is called a polynomial with the Hurwitz property [11, 12], or a stable polynomial if pz( ) 0 on d . Any homogeneous stable polynomial is called a Hurwitz form. An example of the Hurwitz form is the nonzero polynomial
T p( z ) det ( z11 A ... zdd A ) , AAAj j j 0, jd1, , . (2.2) There exist the Hurwitz forms that cannot be represented in the form (2.2) (see [11]).
mk The polynomial ring is factorial. Than the Hurwitz form is a product p( z ) [ pk ( z )] of the irreducible Hurwitz forms. Each irreducible Hurwitz form pzk () is indefinite, that is, it changes sign to d .
3. Positive Real Functions and Degree Reduction Operator
The main object of research in the study of positive real functions is matrix forms of a special type and the possibility of representing such forms as a sum of squares of forms.
mm 3.1. Proposition. If f()()/() z P z q z d , then partial Wronskians P()() z q z W[(),()] q z P z q () z P () z , kd1, , (3.1) zk zzkk are PSD forms.
Proof. By the condition f( z ) f ( z ) 0 on d , we put i . Since f()() z f z , \{0}, it is observed that f()() z f z f()() z f z 0 , Imz 0 ; 0 , Imz 0 , . (3.2) 2i k 2i k ˆ Suppose k 1 and xx2,,d ; then univariate function f()(,,,) f x2 xd satisfies the inequality dfˆ( ) / d 0 . From here ˆ P()() x q x2 df d q( x ) P ( x ) q ( x ) ( x1 ) 0 , где x(,,) x12 x xd .■ z11 z d
mm The following statement allows us to eliminate functions f()()/() z P z q z d satisfying the condition degP ( z ) deg q ( z ) . (3.3) zzkk
3.2. Proposition. Suppose satisfies the assumption of (3.3); then there exist a positive semidefinite real mm matrix Ak and a form Pz1() such that: (a) degP ( z ) deg q ( z ); (b) f()()/() z P z q z mm ; (c) f()() z z A f z . zzkk1 11 d kk 1
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Proof. The degrees of the numerator and denominator of a positive real function fz() for each variable cannot differ by more than 1. Suppose k 1; then there exists a limit fz() limA12 ( z , , zd ) 0 , z 1 z1 d1 where the matrix-valued function A12(,,) z zd is holomorphic on . d1 ˆ For any zˆ (,,) z2 zd the matrix-values function f()(,) z11 f z zˆ satisfies the ˆˆ ˆ inequality f( z11 ) f ( z ) 0 , Rez1 0 . The function fz()1 has only simple poles on the imaginary axis (including the pole at ) and the residues at these poles are positive semidefinite matrices. Then ˆ ˆ fz()1 d1 res f ( z11 ) lim A ( zˆ ) 0 for all zˆ . z z 1 1 z1
d1 Since Az1()ˆ is holomorphic in , we see that A11( zˆ ) A 0 , where A1 is a constant matrix. The function satisfies the Schwarz-Pick inequality
fˆ()()()() z f ˆ z f ˆ z f ˆ x 1 1 1 1 z z z x 1 1 1 1 0 , (3.4) ˆ ˆ ˆ ˆ f()()()() z1 f x 1 f x 1 f x 1 z1 x 1 x 1 x 1 where Rez1 0 and x1 0 . Limit of (3.4), when x1 is tending to , is a matrix
f()() z f z A1 zz11 0. (3.5) AA11 Therefore, we have f()() z f z f()() z f z A1 Im AII1 mm zz11 0 . (3.6) I zz11 m AA11
It follows from (3.6) that function f1()() z f z A 1 z 1 satisfies the inequality f11( z ) f ( z ) 0, d mm z . Since f11()() z f z , \{0}, we see that fz1() d .■ Further simplification is based on the use of the degree reduction operator ([19], [12]). In some cases, this allows us to restrict ourselves in considering multi-affine rational positive real functions. An example of a multi-affine k -form is an elementary symmetric polynomial of degree in variables 1,, n : (,,) , kn1, , ; ( , , ) 1. (3.7) k1 n i12 i ik 01 n i12 i ik n n! Polynomial (3.7) is the sum of monomials. k k!( n k )!
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Let p(,) z z be an l -form, where (,)(,,,)z z z z z d1. If degp ( z , z ) n , then 0 0 0 1 d z0 00
n0 k p(,)() z00 z plk z z , (3.8) k0 where pz() are ()lk -forms in variables zz,, . Let, further, ,, be new variables. lk 1 d 1 n0 3.3. Definition. Transformation
1 nn00n n0 k 0 D :p () z z p () z (,,) , (3.9) z0 l k01 l k k n0 kk00k is called a degree reduction operator in the variable z0 .□ If f(,)(,)/(,) z z p z z q z z , where degf ( z , z ) n , then by definition, put 0 0 0 z0 00 n0 D p(,) z0 z n0 z0 ˆ D f(,):(,,,) z01 z f n z .□ (3.10) z0 Dn0 q(,) z z 0 z0 0
Note that the degrees of p(,) z0 z and q(,) z0 z in the variable may differ, but the same degree reduction operator Dn0 is applied to the numerator and denominator. z0 1 n Since 0 (,,) zk , we see that transformations (3.9) and (3.10) are kn100 z k 10n0 invertible. It turns out that the degree reduction operator (3.10) preserves the positive reality of the function. mm 3.4. Theorem. If f(,)(,)/(,) z0 z P z 0 z q z 0 z d 1 , where the polynomials P(,) z0 z , q(,) z z are coprime and degf ( z , z ) n , then 0 z0 00 Dn0 P(,) z z ˆ z0 0 fz(,,,)1 n 0 Dn0 q(,) z z z0 0 is the function of class mm , affine in each of the variables ,, . dn 0 1 n0 We will need a few auxiliary statements. 3.5. Proposition. The coprime numerator and denominator of a scalar positive real function are the Hurwitz forms. Proof. The homogeneity of the polynomials is obvious. Stability easily follows from a similar fact for functions of one variable having a nonnegative real part in the right half-plane.■ 3.6. Lemma. Let p(,) z z , z d be a Hurwitz form, where degp ( z , z ) n . Then the 0 z0 00 polynomial pˆ( , , , z ) Dn0 [ p ( z , z )] is also a Hurwitz form. 10n0 z0
d Proof. If (,,)zzˆˆ1 d , than a polynomial p(,) z0 zˆ in one variable z0 has no zeros in the right half-plane. We apply the degree reduction operator in the variable . By the Grace-Walsh-
Szego Theorem, there exist a point , Re 0 such that pˆˆ(,,,)(,,,)1 n zˆˆ p z
pz( ,ˆ ) 0. The homogeneity of the polynomial pzˆ(,,,)1 n ˆ is obvious.■
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Proof of Theorem 3.4. A matrix-valued function f(,) z0 z is positive real if and only if for any T real vector the scalar function f(,) z0 z is positive real. Since addition does not deduce from the class of positive functions, we see that
p(,)(,)(,) z0 z zd 1 q z 0 z p z 0 z zd1 q(,)(,) z00 z q z z is a positive real function in variables z0,,,, z 1 zdd z 1. Its numerator is Hurwitz form. The degree reduction operator is additive. By Lemma 3.6, the polynomial DDDn0[zqzzpzz (,) (,)] z n 0 [(,)] qzz n 0 [(,)] pzz (3.11) z0dd1 0 0 1 z 0 0 z 0 0 is the Hurwitz form in variables ,,,,,,z z z . Polynomial (3.11) vanishes at the point 1n0 1 d d 1 z0 DDnn00[( p z ,)]/ z [( q z ,)] z . Since the polynomial (3.11) is stable, we see that d1zz00 0 0 Rez0 0 , (,,,,,)zz dn 0 . Hence ReDDnn00 [(,)]/p z z [(,)] q z z 0 , d1 11nd0 zz0000 .■ 3.7. Theorem (criterion of positivity). A rational multiaffine real homogeneous f( z ) f ( z ), \{0} matrix-values function f()()/() z P z q z belongs to the class P()() z q z mm iff all Wronskians W[,]()() q P q z P z , kd1, , are PSD forms. d zk zzkk Proof. The necessity is proved in Statement 3.1. Let us prove the sufficiency. Since fz() is multiaffine, we see that
Pz() zk P12()() zˆˆ P z f(,) zk zˆ , q()()() z zk q12 zˆˆ q z d1 where zˆ (,,,,,) z1 zk 1 z k 1 z d . If zxˆ ˆ , than
1 zkk P1()()()() xˆ P 2 x ˆ z P 1 x ˆ P 2 x ˆ Imf ( zk , xˆ ) 2izqxkk1 (ˆ ) qx 2 ( ˆ ) zqx 1 ( ˆ ) qx 2 ( ˆ ) z zP()()()() xˆ q x ˆ P x ˆ q x ˆ Im z k k1 2 2 1 k W[ q , P ]( xˆ ) . 2i 22zk zkk q1()()()() xˆ q 2 x ˆ z q 1 x ˆ q 2 x ˆ
Hence Imf ( zk , xˆ ) 0 , Imzk 0, for each (for any real other variables). Therefore (see [5], theorem 2.4) Imfz ( ) 0 for zid . Since is homogeneity, we see that mm fz() d .■ 3.8. Remark. If all Wronskians W[,] q P are PSD forms, and the function f()()/() z P z q z zk is not multiaffine, then, generally speaking, it is impossible to guarantee that fz() belongs to mm the class d .
4. Artin's Denominators of PSD not SOS Forms
Let F()[,,] z z1 zd be a PSD not SOS form. By Artin's theorem, there exists a form sz() such that s()() z2 F z is a SOS form. The form is called Artin's denominator of Fz().
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4.1. Proposition. Suppose s()() z2 F z is a SOS form and each irreducible factor of sz() is an indefinite form; then Fz() is also a SOS form. Proof. Assume the converse. Let Fz() be a PSD not SOS form. We decompose the form
into irreducible factors s()()() z s1 z sm z , where in the product some factors can occur several times. Consider the forms
2 2 2 Fz1() szFzFz 1 ()(), 2 () szFz 2 () 1 (), , Fzm () szF m () m 1 () z . (4.1)
Successively applying Proposition 2.4 to the forms F( z ), F11 ( z ), , Fm ( z ) , we obtain that the 2 form Fm ()()() z s z F z is a PSD not SOS form. Contradiction.■ 4.2. Definition. Artin's denominator sz() of a PSD not SOS form is called a minimal
Artin denominator if a form sˆ()()/() z s z sj z is not Artin's denominator of the form for each of irreducible factor szj () of .□
4.3. Theorem. Let , z d be a PSD not SOS form. Then: (a). There exists a minimal Artin denominator sz() of ;
(b). Each irreducible factor of does not change sign to d .
Proof. By Artin's theorem, there exists a form rz() such that r()() z2 F z there is a SOS form. Each irreducible factor of the form is either indefinite or does not change sign on . Then d r()()() z s0 z s z , where all irreducible factors of the form sz () do not change sign on , and 2 the irreducible factors of the form sz0 () are indefinite. Consider the form F1()()() z s z F z . 22 By condition, s01()()()() z F z r z F z is a SOS form. The irreducible factors of the form are indefinite. Then, by Proposition 4.1, sz () is also Artin's denominator of the form . Removing all "excess" irreducible factors from the form , we obtain Artin's denominator with the required properties.■
5. Product Polarization Theorem
We need a special representation for the product of two forms. Such a representation is a certain "modification" of the long-resolvent Theorem. 5.1. Theorem (Product Polarization Theorem). Suppose qz() is a real form of degree n , and P()() zT P z is a real mm matrix-valued form of degree n 1, where z d ; then:
(i). There exist real symmetric matrices Ak , kd1, , such that
T d q()() P z ()( z11 A ... zdd A )() z , , z , (5.1)
1 N j where ()(,,)z z Imm z I . Here Im is the matrix unit, and z , jN1, , are all monomials of degree in the variables zz1,, d ; (ii). The partial Wronskians W[ q ( z ), P ( z )] are represented as zk
P()() z q z T q()()()() z P z z Ak z , . (5.2) zzkk
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The proof of Theorem 5.1 is based on the following statement. n! 5.2. Lemma. Let z12,,, z zN , N be all multiaffine monomials of degree n d!( n d )! in variables z1, , zd , 1 n d 1. For any multiaffine monomial z of degree n 1 there exists a linear pencil C() z z11 C zdd C whose coefficients Cj ( j 1, , d ) are real symmetric NN matrices of rank no more than 2 such that
z 1 z z2 0 ()z C z C . (5.3) 11 dd z N 0
Proof. We split a set of variables in z , z1 into three disjoint subsets: (i). Common variables of monomials z1 and z . These variables form the largest common 1 divisor z of monomials z , z ; (ii). Variables in not belonging to z1 . Denote these variables by odd indices: 1 z1,,, z 3 z 2k 1 . Since degzz deg 1, we see that k 1; (iii). Variables in not belonging to . Denote these variables by even indices: z2,,, z 4 z 2k 2 . If k 1, then this subset is empty. We have
1 z zzzzz246 2422k k z; z zzzzz 135 2321 k k z .
11 (a). If k 1, then z z; z z11 z z z , then
1 z1 00z z 0 0 0 z2 0 , 0 0 0 N z 0 i.e., C() z z11 C , where C1 is a real symmetric matrix of rank 1. (b). If k 2 , then we put
1 2 z z2 z 4 z 6 z 2kk 4 z 2 2 z ; z z3 z 5 z 2kk 3 z 2 1 z .
The monomials z j , jk3,4, ,2 1 are defined by the formulas z zzjj2 j2 , jk3,4, ,2 1. z j1 If ij , then zzi j and
2 jj2 22k 1 21k z1 z z , zjj z z1 z , jk1, ,2 3 , z2kk 2 z z 2 1 z , z21k z z . (5.4) From (5.4) follows the identity
13
1 z 0zz1 0 0 0 0 0 2k 1 z z2 zz120 0 0 0 0 0 0 0zz 0 0 0 0 0 z 3 0 23 4 0 0zz34 0 0 0 0 z 0 1 . 0 0 0z4 0 0 0 0 z 5 0 2 0 0 0 0 0 0z 0 23k 0 23k z 0 0 0 0 0zz2kk 3 0 2 2 0 z 22k zz0 0 0 0 0 0 2kk 1 2 2 0 z 21k We have linear pencil
C() z z1 C 1 z 2 C 2 z 2kk 1 C 2 1 whose coefficients C j , jk1, ,2 1 are a real symmetric matrices of rank 2 . Supplementing the linear pencil Cz() with zeros to the required size, we obtain (5.3).■
Proof of Theorem 5.1. Let qz(), Pz() be the forms satisfy the conditions max{degq ( z ),deg P ( z )} n , kd1, , . zkk z k
Applying the degree reduction operators Dnk , to the forms qz() and Pz(), we obtain zk the multiaffine forms that satisfy the conditions of the theorem. If Theorem 5.1 is true for the multiaffine forms, then after identifying the corresponding groups of new variables with the variables zk , , we obtain relation (5.1) for the original forms. Therefore, it suffices to prove Theorem 5.1 for the multiaffine forms.
N j l k Let q() z j1 aj z , P() z k1 Bk z be the multiaffine forms of degree n and n 1. By Lemma 5.2, for a fixed monomial z j , jN1, , , and each monomial zk , kl1, , , there exists a symmetric real matrix pencil Czjk () such that
z1 0 j1 z 0 C( z ) j k j - th row , , . jk z z z j1 0 N 0 z
From the coefficients a j , Bk of forms qz(), Pz() and pencils Czjk () we construct the matrix pencil Nl A( z ) z11 A ... zd A d a j C jk ( z ) B k , (5.5) jk11
14
N where C B {} cjk B j,1 k is the Kronecker product of the NN matrix C and the mm matrix B . The matrix pencil (5.5) satisfies the condition
1 z Im a1 P() z Az() . (5.6) N a P() z zIm N The properties of reality and symmetry of Az() are obvious. Multiplying (5.6) on the left by the
1 N d matrix (,,)IImm, , we obtain relation (5.1) for the multiaffine forms.■
Note that the matrix pencil z11 A... zdd A is not uniquely determined by the forms qz(), Pz(). An easy analysis shows that the following statement holds. 5.3. Proposition. Under the conditions of Theorem 5.1, the difference of any two matrix pencils representing the product of forms q()() P z is a matrix pencil z11 S... zdd S such that
T T (z11 S ... zdd S ) ( z ) 0, (z ) Sk ( z ) 0 .□ (5.7)
6. Gram's Matrices and Representation Defect Lemma
Let Fz(), z d be a real 2n -form such that degF ( z ) 2 n , kd1, , , and let zkk
j N j {} z be the set of monomials of degree n satisfying the condition deg z n n , j1 zkk .
Suppose the form has two Gram's matrices A1 , A2 ; then for the real matrix SAA12 the relation holds s s z1 11 1N 1 N (zz , , ) 0 , sij s ij s ji . (6.1) ssN1 NN z N
The set of matrices S satisfying (6.1) is a linear space L0 . Before considering the Representation Defect Lemma, we construct a special basis in the linear space .
6.1. Proposition. In the linear space L0 of real symmetric matrices satisfying condition (6.1) there exists a basis such that each basis matrix has either 3 or 4 nonzero elements at the intersection of rows and columns corresponding of monomials
22 zr z, z r z l z , z l z , (6.2)
1 1 2 2 12 zr z; z l z ; z l z ; z r z , zz . (6.3) The nonzero submatrices of the corresponding basis matrices are determined by the relations 2 0 0 1 zzr 22 (zr z , z r z l z , z l z )020 z r z l z 0 , (6.4) 2 1 0 0 zzl
15
1 0 0 1 0 zzr 0 0 0 1 zz1 1 1 2 2 l (zr z , z l z , z l z , z r z ) 0 . (6.5) 1 0 0 0 zz 2 l 0 1 0 0 2 zzr 6.2. Comment. Relation (6.1) can be conveniently rewritten as
N ik j F( z ) sij z z c k z 0 , (6.6) i,1 j k where degznj 2 , kd1, , , and zzi j , ij . zkk
It follows from (6.6) that the elements sij of the matrix (6.1) satisfy the conditions
sij 0.□ (6.7) i j k We will need several lemmas. i j i j i j Let Π be the set of all unordered pairs (,)zz, zz, such that z z z ,
r1 rd where z z1 zd is a fixed monomial of degree 2n .
1 d 6.3. Lemma. Let (,,)rr1 d be a multi-index. Monomial z z1 zd belongs to some pair Π iff
max{(rk n k ),0} k min{ r k , n k }, kd1, , . (6.8)
1 11dd1 Proof. If z z z , where z z11 zdd, z z z , then vrk k k . Hence
0 kkn , rk n k k r k . These inequalities coincide with (6.8). Sufficiency is obvious.■
1 r ld 6.4. Definition. If a monomial z z1 zr zld z satisfies conditions rr n ,
l 1, then a transformation
11rrl d1 l1 d z11 zrr zl z d z z z l z d (6.9) is called an elementary monomial transformation.□
6.5. Lemma. Suppose monomials zzi , j , zzi j satisfies the assumption (6.8); then: (i). There exists a chain of pairwise different monomials satisfying conditions (6.8) "connecting" monomials zi and z j ; (ii). Any two neighboring monomials of the chain are related by an elementary transformation.
i 1 d j 1 d Proof. Let z z1 zd and z z1 zd be two different monomials satisfying condition (6.8). If k j j j , jd1, , , then d min{rj , n j } k j min{ r j , n j } and k j 0 . j1
16
We represent an integer tuple (,,)kk1 d in the form of a componentwise sum of the elementary tuples. Each elementary tuple (,,)mm1 d contains only two nonzero components 1 and 1. The numbers +1 are associated with k j 0 , and with k j 0. The sequence of the elementary tuples corresponds to the sequence of elementary transformations of
i 1 d monomials. We choose as the initial monomial the monomial z z1 zd . At each step we obtain a monomial that satisfies conditions (6.8) and does not coincide with the monomials
j 1 d obtained in the previous steps. At the last step, we get the monomial z z1 zd .■
We say that elements 12, Π are connected by an elementary transformation if one of the monomials of the pair 1 is connected by an elementary transformation with one of the monomials of the pair 2 . kv Note that any element of (,)zzΠ is uniquely determined by the choice of one of the monomials of the pair.
6.6. Corollary. If 12, Π and 12 , then:
(i). The set Π contains a chain of elements that "connects" 1 and 2 ; (ii). Any two neighboring elements of the chain are related by an elementary transformation. Proof. It suffices to consider the chain connecting the first monomials from , .
6.7. Lemma. If Π contain m 2 elements, then in the set there exist (m 1) different pairs {,}ij such that the elements of each pair are connected by an elementary transformation. Proof. Let us assign a finite graph to the set . The vertices of the graph are elements of the set . The edges of the graph form pairs of elements connected by an elementary transformation. By Corollary 6.6, the graph is connected. The graph tree (that is, a connected subgraph with the same vertices, but without cycles) contains the number of edges one less than the number of vertices. Since m 2 and in the graph tree different edges are incident to different pairs of vertices, the number of different pairs connected by an elementary transformation is m 1.■ Proof of Proposition 6.1 . If each Π contains at most one element, then dimL 0 . k 0
Suppose some of the sets contain elements; then m 1 coefficients sij , ij in the sum (6.7) can be chosen arbitrary, and the remaining coefficient is determined from the condition ck 0 . Thus, such set defines a subspace of dimension of the space L0 . Let us show that in each such subspace there exists a basis formed by the matrices of the form (6.4) and (6.5).
By Lemma 6.7, in there exist different pairs of elements such that i and
j are connected by an elementary transformation. The following cases are possible:
ii (a). One of the elements of a pair {,}12, say 1 , has the form 1 (,)zz, that is ik2 j l i j ()zz . Let the second element of the pair be 2 (,)zz and the monomials z , z are connected by an elementary transformation. Then there exist a variables zzrl, such that
17
j 1 r 1 ld1 i 1 r ldl 1 r 1 ld1 z z1 zr zld z , z z1 zr zld z , z z1 zr zld z , or j 2 i l 2 1 r 1 ld1 z zr z , z zrl z z , z zl z , z z1 zr zld z . This triplet of monomials corresponds to the basis matrix (6.4).
i j i j ls ls i l (b). 1 (,)zz , zz ; 2 (,)zz , zz , and monomials z , z are connected by an elementary transformation. Then there exist variables zzrl, such that
i 1 r ldl 1 r 1 ld1 z z1 zr zld z , z z1 zr zld z ,
j 11 rr l l d d s 11 rr1 l l 1 d d z z1 zr zld z , z z1 zr zld z , or
i 1 l 1 j 2 s 2 z zr z , z zl z , z zl z , z zr z , (6.9)
1 1 r 1 ld2 r11 rrr rrl l 1 d d where z z1 zr zld z , z z1 zr zld z . 12 12 In addition zz . Otherwise, if zz , then 12 . Contradiction. The monomials (6.9) define the basis matrix (6.5).
Since all pairs {,}ij are different, then corresponding (m 1) matrices are linearly independent.■
6.8. Representation Defect Lemma. Let z j , jN1, , be all monomials of degree n in variables zz,, such that deg znj , kd1, , . 1 d zkk
If a symmetric real matrix Sd satisfy the conditions
z1 nd zz1 / nd d 1 N (z , z ) Sd 0 , Sd 0, (6.10) N ndN nd z zz/ d then there exist the symmetric real matrices Sk , kd1, ,( 1) for which
z1 0 (z1 S 1 ... zd 1 S d 1 z d S d ) . (6.11) z N 0 Proof. Without loss of generality, we can assume that
1 N nd TT (z ,, z )( zd (), zˆˆ (,)) z zd , where deg (zˆ , z ) n 1, and the components of ()zˆ are different monomials in the zd d d variables (,,)z11 zd zˆ . Dividing the matrix Sd into blocks, from the second relation (6.10) we obtain
SS11 12 nz!() ˆ d 0 . T ˆ SS12 d 0
18
00 T T ˆ Hence S 0 , S 0 . Then S and (zˆˆ , z ) S ( z , z )) 0 . 11 12 d ˆ d d d 0 Sd We rewrite (6.11) in block form
n A11()() zˆˆ A 12 z zzd () 0 d . (6.12) T ˆ A12()() zˆˆ A 22 z zdd S (,)zzˆ d 0 ˆ ˆ Matrix Sd is a linear combination of basis matrices Sdj, of the form (6.4), (6.5). If there exists a solution A11( zˆ ), A 12 (z ˆ ), A 22 (z ˆ ) of equation (6.12) for each basis matrix , then the general solution (6.12) will be a linear combination of basic solutions.
2 2 The basis matrix (6.4) is determined by the monomials zzr , zrl z z , zzl . By construction, we have deg (zˆ , z ) n 1. Among the components of the vector (znd TT ( zˆˆ ), ( z , z )) zd d d d d there exist monomials zdr z z and zdl z z . The solution of the equation (6.12) for the basis matrix (6.4) is the matrix 0 0 0 zz zdr z z lr z z z 0 0zzlr 0 dl 2 0zzld 0 0 zz 0 . r zl z r0 2 z d 0 z z z rl zzrd0 0 0 2 zzl
1 1 2 2 For the monomials zzr , zzl , zzl , zzr defining the basis matrix (6.5), there exist 1 2 monomials zzd and zzd such that
2 zzd 0 0zzvk 0 0 1 0 0 0 0 zzzzd kv z0 0 0 z 0 z z1 v d k 0 . zzkd0 0 0 0 zz1 v 0zzkd 0 0 0 zz 2 v 0zzvd 0 0 0 2 zzk The general solution of equation (6.12) is a linear combination of such solutions.■
7. Rational Functions with PSD not SOS Wronskian
It follows from Theorem 5.1 that rational function f()()/() z p z q z is represented as
p()()() z z T f( z ) ( z A z A ... z A ) , , z d . (7.1) q()()() z q 1 1 2 2 dd q z Let be a rational function with PSD partial Wronskian W[,] q p . Let us zk investigate the possibility of obtaining a representation of the form (7.1) with a positive semidefinite matrix Ak 0.
19
7.1. Theorem. Let qz(), pz(), z d be real forms, where degp ( z ) 1 deg q ( z ) , degp ( z ) deg q ( z ) . If partial Wronskian W[,] q p is a PSD not SOS form with Artin's zz11 z1 denominator sz(), then:
(i). There exists a matrix pencil A( z ) z1 A 1 z 2 A 2 ... zdd A with a positive semidefinite matrix A 0 such that 1 p()()() z z T f( z ) ( z A z A ... z A ) , , z d , (7.2) q()()()()() z q s1 1 2 2 dd q z s z
j where ()z is a row vector of monomials z in the variables zz1,, d satisfying the conditions degz j deg q ( z ) s ( z ) , deg (z ) deg q ( z ) s ( z ) n ; zz111 (ii). Partial Wronskians are represented as ()()zzT W[ q ( z ), p ( z )] A , kd1, , . (7.3) kks()() z s z
Proof. By Theorem 5.1, there exists a real symmetric matrix pencil z1 B 1 z 2 B 2 ... zdd B such that T qspzsz()()()() ()[ zBzB1 1 2 2 ... zBdd ]() z , , (7.4)
T W[,]()() qs ps z B z , kd1, , . (7.5) zkk
Differentiating (7.4) n1 1 times with respect to the variable z1 , we obtain
n111p()()()() z s z n 1 zTT n 1 z q()() s (1)() n B ()( z B ... z B ) . (7.6) n1111 1 n 1 1 1 dd n 1 z1 z 1 z 1
n11 p()() z s z n11 ()z T By condition, we have 0 , 0 . Using (7.6), we obtain n11 n11 z1 z1 n1 ()z T B 0 . (7.7) 1 n1 z1 The form
T ()()ps qs22 p q ()()[,][,]zB z Wqspsqs ps sq p sWqp 1 zz11 z1 z 1 z 1 z 1 is SOS form. The matrix B1 is its Gram matrix. Among the Gram matrices of the SOS form, there exists a positive semidefinite matrix A1 0:
TT Wqsps[,]()()()() zB z zA z . z1 11 The degree of the SOS form W[,] qs ps (and hence PSD) with respect to the variable z is z1 1 necessarily even. Then T deg ()z A () z deg W [,]2 qs ps n 2. (7.8) z111 z 1 z 1
It follows from (7.8) that diagonal elements a jj of the matrix A1 corresponding to the
j T n1 monomials z of the vector ()z containing z1 are necessarily equal to 0 . From the positive semidefiniteness of A1 we obtain
20
n1 ()z T A 0 . (7.9) 1 n1 z1
T By (7.9) and (7.7) it follows that matrix SAB1 1 1 , SSS1 1 1 satisfies the conditions
n1 T T ()z (z ) S ( z ) 0 , S 0 . 1 1 n1 z1 Using Representation Defect Lemma, we get
T d ( )(z1 S 1 z 2 S 2 ... zdd S ) ( z ) 0 , , z , (7.10) where Sk , kd2, , are the real symmetric matrices. Adding (7.10) to (7.4) and dividing both sides of the resulting identity by the product q()()()() s q z s z , we obtain (7.2).
The relations (7.3) follow from the identities sWqp2 [,][,]()() Wqsps zA z T .■ zkk z k
8. Sum of Squares Theorem
d d d By i { z : Im zk 0, k 1,,} d denote the open upper poli-halfplane. The d d d closure of is {z : Im zk 0, k 1,,} d . Tо prove the Sum of Squares Theorem, we need the following statement.
8.1. Proposition. Let p( z ), q ( z ) [ z1 , , zd ] , d 3 be are forms such that: (i). degp ( z ) 1 deg q ( z ) ; (ii). degp ( z ) deg q ( z ) ; zz11 p()() z q z (iii). The partial Wronskian W[,]()() q p q z p z is a PSD not SOS form. z1 zz11 d Then f()()/() z p z q z has a singularity at some point (,,)z12 z zd , Imz1 0 . First, we prove
8.2. Лемма. Let s01()[,,] z z zd , s01( z ) / z 0 be a irreducible form that does not d change sign to . If the form sz0 () is not a divisor of the polynomial h()[,,] z z1 zd , then there exists a point z(,,,)12 x xd , Im1 0, xx2,,d such that sz0 ( ) 0, hz( ) 0 . Proof. Each irreducible form that does not change sign on d depends on at least two variables. Since s( z ) / z 0, we see that degsz ( ) 2 and there exists a variable z , k 1 01 z1 0 k for which s0 ( z ) / zk 0.
The form is not a divisor of the polynomial hz(). Then there exist xx2,,d ,
0 such that for any xx2,,d satisfying the condition xxkk , kd2, , , at least one zero of the polynomial s0(,,,) z 1 x 2 xd in the variable z1 is not a zero of the polynomial h(,,,) z12 x xd . The real zeros of s0(,,,) z 1 x 2 xd have even multiplicity, and the remaining zeros form complex conjugate pairs. For an arbitrarily small change in the numbers (and hence the coefficients of the polynomial), all real multiple zeros split into
21
complex conjugate pairs. Then there exists 1 , Im1 0 such that s0( 1 , x 2 , , xd ) 0 and h(12 , x , , xd ) 0 .■ Proof of Proposition 8.1. By Theorem 4.3, for PSD not SOS form W[,] q p , there exists a z1 k1 kn minimal Artin's denominator s()()() z s1 z sn z with irreducible factors szj (), jn1, , that do not change sign to d . There are two alternative possibilities for sz():
(a). There exists an irreducible factor szj () depending on the variable z1 ; (b). Artin's denominator sz() does not depend on the variable .
By Theorem 7.1, there exists a symmetric matrix pencil A( z ) z1 A 1 z 2 A 2 ... zdd A with a positive semidefinite matrix A1 0 such that
p()()() z z T f( z ) ( z A z A ... z A ) . q()()()()() z q s1 1 2 2 dd q z s z Hence we get
d d Imf ( z ) Im zkk ( z , z ) , Ref ( z ) Re zkk ( z , z ) , (8.1) k1 k1 T H()() z H z Wz [ q ( z ), p ( z )] , (8.2) 1 s()() z s z
1/2 where H() z () z A11 ((),, h z hN ()) z and
1H ( z ) H ( z ) 1 (zz ) ( ) 1(,)zz 2 ; kk(,)z z2 A , kd2, , . (8.3) qz() s()() z s z qz() s()() z s z The following observation is essential. In relation (8.2), the right side is a polynomial (we cancel sz()2 out of that fraction). If z d , then the expressions
H()() z H z ()()zz and Ak s()() z s z s()() z s z
d 2 in (8.3) are still polynomials, but if z , then square of the modulus sz() no longer "cancels out". Then the complex zeros of sz() can be singularity points for the functions Refz ( ) and Imfz ( ) in (8.1).
Each irreducible factor szj () cannot be a divisor of all elements of the row vector Hz(). If
is a divisor of all elements , then it follows from (8.2) that sˆ()()/() z s z sj z is also Artin's denominator of the Wronskian W[ q ( z ), p ( z )]. This contradicts the minimality of sz(). z1 Let hzˆ() denote an element of for which is not a divisor.
T Since AAAkkk, , we see that there exist real matrices Rk (generally rectangular) such that ARJRk k k k , where Jk are diagonal matrices with elements 1 on the main diagonal. Let us introduce the designate (z ) R ( g()()kk ( z ), , g ( z )) . Then k 1 rk
22
2 ()k 2 N hz() rk gz() j ()k j ()k 1(,)zz 2 ; kj(,)zz 2 , j 1, kd2, , . (8.4) j1 q()() z s z j1 q()() z s z
ˆ Case (a). Let sz0 (), s01( z ) / z 0 not be a divisor of an element hz() from Hz(). By d Lemma 8.2, there exists a point zx(,)1 ˆ , Im1 0 such that s( z ) s0 ( z ) 0 , hzˆ( ) 0 . It follows from (8.1) that 2 N h(,) z xˆ ˆ j 1 . Imf ( z1 , x ) Im z 1 1 Im z 1 2 j1 q(,)(,) z11 xˆˆ s z x
We obtain lim Imf ( z1 , xˆ ) . That is, the function fz() has a singularity at the point z11 d (,,,)12xx d , Im1 0.
Case (b). Let Artin's denominator s()(,,,) z s z23 z zd be independent of the variable z1 .
Without loss of generality, we can assume that s02( z ) / z 0 , where s02(,,) z zd is an irreducible factor of sz(). Let not be a divisor of an element from . By Lemma d1 ˆ 8.2, there exists zxˆ(,)2 ˆ , Im2 0 such that sz0 (ˆ ) 0. We have h( z12 , , xˆ ) 0 for almost all z1 , Imz1 0 except for a finite number of points.
Without loss of generality, we can assume that Re2 0 . If Re2 0 , then instead of the variable z2 consider a new variable zˆ2 z 2 x 2 , x2 \{0}.
Let us show that for all z1 , , the double limit 2 N 1 hj (,,) z12 z xˆ lim1 (z 1 , z 2 , xˆ ) lim (8.5) z z z z 2 1 1 1 1 q(,,) z z xˆ j1 s(,) z2 xˆ zz2 2 2 2 12 either does not exist or is equal . ˆ For each z1 , Imz1 0 there are two possibilities: either h( z12 , , xˆ ) 0 , or . In the first case, by Theorem 2.2, the limit (8.5) does not exist, and in the second case (8.5) is equal to . Further reasoning is based on the relations
Im(,,)Imfzzx1 2ˆ z 1 1 (,,)Im zzx 1 2 ˆ z 2 2 (,,) zzx 1 2 ˆ , (8.6)
d Re(,,)Refzzx12ˆ z 1112 (,,)Re zzx ˆ z 2212 (,,) zzx ˆ xkk (,,) zzx 12 ˆ . (8.7) k3 d There are two alternative possibilities for the sum xkk (,,) z12 z xˆ in (8.7): k3
(b.1). For each z1 , Imz1 0 , Rez1 0 double limit d limxkk ( z12 , z , xˆ ) (8.8) zz11 k3 z22 either does not exist or is equal to ;
23
(b.2). There exists a finite limit (8.8) at the point z1 , Imz1 0 , Rez1 0 .
Case (b.1). Limit (8.8) either does not exist or is equal to . For Re2 0 there exists a
Re2 point (,,)zx12 ˆ such that Imz1 0 , Rez1 0 and Rezz11 Im . Im2
Suppose that the function fz() is regular at the point (,,)zx12 ˆ ; then there exist finite limits
limIm(,,)fzzx1 2ˆ limIm z 1 1 (,,)Im zzx 1 2 ˆ z 2 2 (,,) zzx 1 2 ˆ , z1 z 1 z 1 z 1 zz2 2 2 2 d limRe(,,)fzzx12ˆ limRe z 1112 (,,)Re zzx ˆ z 2212 (,,) zzx ˆ xkk (,,) zzx 12 ˆ . z1 z 1 z 1 z 1 k3 zz2 2 2 2 Since limit (8.8) either does not exist or is equal to , we see that
lim Rez1 1 ( z 1 , z 2 , xˆˆ ) Re z 2 2 ( z 1 , z 2 , x ) zz11 z22 must have the same property. From condition Rez11 K Im z , where K (Re22 / Im ) 0 we obtain
limIm(,,)fzzx1 2ˆ limIm z 1 1 (,,)Im zzx 1 2 ˆ z 2 2 (,,) zzx 1 2 ˆ z1 z 1 z 1 z 1 zz2 2 2 2
(1/)limReK z1 1 (,,)Re z 1 z 2 xˆˆ z 2 2 (,,) z 1 z 2 x . zz11 z22 The last limit either does not exist or is equal , which contradicts the assumption. The d function fz() has a singularity at the point (,,)zx12 ˆ , Imz1 0 .
Case (b.2). Let us prove that f(,,) z12 z xˆ also has a singularity at the point (,,)zx12 ˆ .
Assume the converse. Then function fxˆ (,)(,,) z1 z 2 f z 1 z 2 xˆ is continuous at the point
(,)z12 , i.e., there exist finite limits
lim Ref ( z12 , z , xˆ ) , lim Imf ( z12 , z , xˆ ) . zz11 zz11 z22 z22
Then rational function fxˆ (,) z12 z is continuous in a sufficiently small neighborhood of the point d . If limxkk ( z12 , z , xˆ ) A , then (see (8.7), (8.6)) there exist finite limits zz11 k3 z22
lim Rez1 1 ( z 1 , z 2 , xˆˆ ) Re z 2 2 ( z 1 , z 2 , x ) , (8.9) zz11 z22
lim Imz1 1 ( z 1 , z 2 , xˆˆ ) Im z 2 2 ( z 1 , z 2 , x ). (8.10) zz11 z22 Since the double limit (8.5) either does not exist or is equal to , we see that the limits of the first terms in (8.9) and (8.10) will be the same. By assumption, both limits (8.9) and (8.10) are finite. Then the points z1 and 2 must necessarily satisfy the relations
24
Rez11 K Im z , Re22K Im , (8.11) where K Re22 / Im 0 .
Condition (8.11) is violated at point (,)z12 , where zz11 , \{0}. For each such point , either limit (8.9) or (8.10) does not exist or is equal to . This contradicts the continuity of the function fxˆ (,) z12 z in a small neighborhood of the point (,)z12 . The function d fz() has a singularity at the point (,,)zx12 ˆ , Imz1 0 .■ 8.3. Corollary. Proposition 8.1 remains valid if replace scalar form pz() by mm matrix form P()() z P z T .
In the proof, one should consider the diagonal elements fzll () of the matrix-valued function f()()/() z P z q z .■
mm 8.4. Sum of Squares Theorem. If P()/() z q z d , then the partial Wronskians P()() z q z W[,]()() q P q z P z , kd1, , (8.12) zk zzkk are SOS forms. Proof. By Proposition 3.1, Wronskians W[,] q P are PSD forms. If d 2 , then each PSD zk mm form is a SOS form. Suppose f()()/() z P z q z d , where d 3. By Proposition 3.2, if degP ( z ) deg q ( z ) , then zzkk
f( z ) z1 A 1 ... zdd A f 1 ( z ) , where f()()/() z P z q z mm , A 0 and degP ( z ) deg q ( z ), kd1, , . 11 d k zzkk1 If W[ q ( z ), P ( z )] are SOS forms and , then Wronskians zk 1 WqP[,] qzA ()2 WqzPz [(),()] zkk k z 1 will also be SOS forms. Let now degP ( z ) deg q ( z ) , . zzkk Assume the converse. Then at least one of the Wronskians, say W[ q ( z ), P ( z )], is a PSD not z1 d SOS form. By Proposition 8.1, there exists a point z (,,,) z12 z zd , Imz1 0 such that mm has a singularity at this point. Since fz() d , we see that at each point d (,,,)z12 z zd , the inequality Imf ( z12 , z , , zd ) 0. Then function fz() is regular at each such point. This contradiction proves the theorem.■
9. Long Resolvent Representations of Positive Real Functions
The following theorem and Corollary 9.2 are the main results of this article.
mm 9.1. Theorem. Let fz() be a rational function of class d . Then: (a). There exists an ()()m l m l matrix pencil
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A11()() z A 12 z A( z ) z11 A ... zdd A (9.1) A21()() z A 22 z with are real positive semidefinite matrices Ak 0, kd1, , ; (b). The function fz() has the long-resolvent representation
1 f()()()()() z A11 z A 12 z A 22 z A 21 z . (9.2)
m m m m 9.2. Corollary. dd for each d 1.□ The proof of Theorem 9.1 is based on the following generalization of Darlington's theorem to the case of a function of several variables. Recall that a rational function P()/() z q z is called multiaffine if degP ( z ) deg q ( z ) 1, kd1, , . zzkk
Pz() z1 P 1()() zˆˆ P 2 z mm 9.3. Proposition. Let fz() d be a multiaffine function, let q()()() z z1 q 1 zˆˆ q 2 z W[,]()() q P zˆˆ T z be a SOS form. If ()zˆ , zˆ (,,) z z has the size mr , then z1 11 1 2 d
P11()() zˆˆ z g11()() zˆˆ g 12 z q11()() zˆˆ q z gz()ˆ (9.3) T T g12()() zˆˆ g 22 z 12()()zˆˆ q z Ir q11()() zˆˆ q z ()()m r m r is a multiaffine function of class d1 , and 1 T fz()()()()() gzgzgzzI11ˆ 12 ˆ 22 ˆ 1r gz 12 ˆ . (9.4) Proof. Representation (9.4) follows from the obvious identity
T z1 P 1()() zˆˆ P 2 z P 1 1 1 fz() 2 . (9.5) z1 q 1()() zˆˆ q 2 z q 1 q1 z 1 q 2/ q 1
()()m r m r Let us prove gz()ˆ d1 . The multiaffinity of gz()ˆ is obvious. According to 2 Theorem 3.7 (the criterion of positivity), it suffices to prove that the forms Fkk()()/ zˆˆ q1 g z z , kd2, , are PSD forms. Since fz() is multiaffine, we have
Pz() z z Pˆ z P ˆ z P ˆ P ˆ fz()1kk 1 1 2 3 4 . (9.6) q() z z1 zkk qˆ 1 z 1 q ˆ 2 z q ˆ 3 q ˆ 4 It follows from (9.6) that
gz()ˆ Pqˆˆˆˆ P q() zˆ F() zˆ q2 1 2 2 1 k , k 1 z T k kr()()zˆ qˆ2 q ˆ 3 q ˆ 1 q ˆ 4 I
1 where kk()()zˆ z qˆ1 q ˆ 2 q ˆ 1 1 does not depend on the variables z1 and zk . zk Let us prove the identity ˆˆ T (Pq1ˆ 2 P 2 q ˆ 1 )( q ˆ 2 q ˆ 3 q ˆ 1 q ˆ 4 ) kk ( zˆˆ ) ( z ) . (9.7) We obtain
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T TT11 k()()()()zˆˆ k z z k qˆ1 q ˆ 2 q ˆ 1 1 z k q ˆ 1 q ˆ 2 q ˆ 1 1 zzkk TT 21 1 2 1TT 1 2 ()()zkk qˆ1 q ˆ 2 z q ˆ 1 q ˆ 1 q ˆ 2 1 1 q ˆ 1 1 1 . (9.8) zk z k z k z k It follows from (9.6) that
T 22fz() ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 11()()()()()()zˆˆ z qz zqPqPkkˆ 3113 ˆ zqPqPqPqP ˆ 41143223 ˆ ˆ ˆ qPqP ˆ 4224 ˆ . z1 Hence T 11T ˆ ˆ ˆ ˆ ˆ ˆ 1 1 2zqPk (ˆ 3113 qP ˆ ) ( qP ˆ 41143223 qP ˆ qP ˆ qP ˆ ) , zzkk T 11 ˆˆ ()qˆˆ3 P 1 q 1 P 3 . zzkk Substituting the last 3 relations in (9.8), we see that identity (9.7) holds.
q zkk qˆ3 q ˆ 4 z q ˆ 3 q ˆ 4 11 Since hz 1 d1 , we see that ()qˆ2 q ˆ 3 q ˆ 1 q ˆ 4 is q/ z1 zkk qˆ 1 q ˆ 2 z q ˆ 1 q ˆ 2 z10 z10 PSD form. Then at xˆ d2 we obtain
1 I() qˆ q ˆ q ˆ q ˆ 00Im 0 Fx(ˆ )mk2 3 1 4 0 .■ k ˆ ˆ ˆ ˆ 1 T 0 Ir 0 (q2 q 3 q 1 q 4 ) Ir ()qˆ2 q ˆ 3 q ˆ 1 q ˆ 4 kr I
1 T 9.4. Lemma. Suppose f()() z g11 g 12 g 22 z 1 Ir g 12 , where 1
g11 g 12 a 11 a 12 a13 1 TT ()d z2 Ir a 13 a 23 . TT 2 g12 g 22 a 12 a 22 a23 Then 1 T A22()() z A 23 z Az12 () f()()()() z A z A z A z , (9.9) 11 12 13 T T A23()() z A 33 z Az13 () where
A11()()() z A 12 z A 13 z a 11 a 12 a 13 0 0 0 0 0 0 TT AzAzAz12()()() 22 23 aaa 12 22 23 z 1 00000 Ir z 2 . 1 TTTT A()()() z A z A z a a d I 0 0 0 00Ir 13 23 33 13 23 r 2 2
Proof. The identity (9.9) is verified by direct computation.
Proof of Theorem 9.1. Suppose f()()/() z P z q z mm , degf ( z ) n , kd1, , . d zkk Without loss of generality (Proposition 3.2), we can consider degP ( z ) deg q ( z ) , zzkk kd1, , . Applying to the function fz() Degree Reduction Operator Dnk in each variable z , zk k kd1, , , we obtain multiaffine function
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ˆ ˆ f(,,)(,,)/(,,)1 n P 1 n qˆ 1 n (9.10) ˆ in variables 1,, n , n n1 ... nd . It follows from Theorem 3.4 that f (,,)1 n is a positive real function. Moreover, degPqˆ (,, )deg ˆ (,, )1 , jn1, , . Then jj11nn P(,,) PP()() ˆˆ fˆ() 1 n 1 1 2 , q(,,)()()1 n 1 q 1 ˆˆ q 2 where q1(ˆ ) 0 . By Theorem 8.4, there exists a multiaffine mr 1 matrix-valued form 1()ˆ such that W[,]()()()()()() q P Pˆ q ˆ P ˆ q ˆ ˆ T ˆ . 1 1 2 2 1 1 1 It follows from Proposition 9.3 that function
P11()()ˆˆ (1) (1) gg()()ˆˆq11()()ˆ q zˆ g(1) ()ˆ 11 12 (1)T (1) T ˆˆ gg12()()ˆˆ 22 12()()q Ir 1 q11()() zˆˆ q z ()()m r m r 11 belongs to the class n1 , and
1 ˆ (1) (1) (1) (1) T f(,,)()()()()1 nr g 11 ˆ g 12 ˆ g 22 ˆ 1 I g 12 ˆ . 1
(1) The matrix-valued function g ()ˆ depends only on (n 1) variables 2,, n and satisfies the conditions of Proposition 9.3. Then (1) (1) (2) (2) (2) g11()()ˆˆ g 12 g 11 g 12 g13 (2)1 (2)TT (2) ()g33 2 Ir g 13 g 23 . (1)TT (1) (2) (2) (2) 2 g12()()ˆˆ g 22 g 12 g 22 g23 From Lemma 9.4 we obtain
1 (2) (2) (2)T gg 1Ir 0 g 22 23 fˆ(,,) g(2) g (2) g (2) 1 12 , 1n 11 12 13 (2)T (2) (2)T g g I 0 2Ir g 23 33 r 2 2 13 where function g(2) g (2) g (2) 11 12 13 (2) (2)T (2) (2) g g12 g 22 Ir g 23 1 (2)TT (2) (2) g13 g 23 g 33 Ir 2 depends only on (n 2) variables 3,, n and satisfies the conditions of Proposition 9.3. Continuing the process, at the (n 1) step we get a positive real matrix-valued function of one variable n : (n 1) ( n 1) ( n 1) g g g AAA 11 12 1n 11 12 1n (n 1) T ( n 1) ( n 1) T g12 g 22 Irn g 2 AAA (n 1) 1 12 22 2n gA()n n n n , (9.11) (n 1) T ( n 1) T ( n 1) TT g g g I AAA12n n nn 12n n nn rn1
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where An 0 is real positive semidefinite matrix. We have long-resolvent representation ˆ 1 T f(,,)()()()()1 n A 11 A 12 A 22 A 11 . A positive real matrix pencil has the form
0 0 0 AAA11 12 1n T AA()()001Ir AAA 11 12 1 12 22 2n . AA()()T n 12 22 00 I TT nr1 n1 AAA12n n nn The degree reduction operator (3.10) is invertible. If the variables of each k -th group are replaced by the variable zk , then for the original positive function f(,,) z1 zd we obtain a long- resolvent representation with positive semidefinite matrices of pencil.■
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