ON A NEW TYPE OF MULTIPLIER SEQUENCES ...

ON A NEW TYPE OF MULTIPLIER SEQUENCES AND DISCRIMINANT AMOEBAE MIKAEL PASSARE, MAURICE ROJAS, AND BORIS SHAPIRO

Abstract. In the classical paper [PS14] I. Schur and G. Pol´ ya introduced and characterized two types of multiplier sequences, i.e. linear operators acting diagonally in the monomial basis of R[x] and sending real-rooted (resp. with all real roots of the same sign) polynomials to real-rooted polynomials. Motivated by a fundamental property of amoebae and discriminants discovered in [GKZ94] we introduce below a new class of sign-independently real-rooted polynomials and describe diagonal operators preserving the latter set. A pleasant circumstance in our description is that this class of diagonal operators essentially coincides with the class of all log-concave sequences.

1. Introduction The theory of linear preservers is a widely developed and active area of mathematics, see [Sur92]. One of its most classical instances is the theory of linear preservers of real-rooted polynomials initiated in the late 19th century by Laguerre and Hermite. Usually, characterization of linear preservers of a given set starts with consideration of linear differential operators with constant coefficients and/or linear operators acting diagonally in an appropriate basis. In the case of preservers of real-rooted polynomials the situation is as follows. Given a sequence of real numbers γ = {γj }, j = 0, 1, 2, . . . consider a linear operator Tγ : R[x] → R[x], acting on each xj by multiplication by γj . We refer to such a Tγ as a diagonal operator corresponding to γ. Following [PS14] we call T a multiplier sequence (”Faktorenfolge”) of the first kind if for any polynomial p(x) ∈ R[x] with all real roots its image T (p) is real-rooted as well. A multiplier sequence of the 2nd kind is a diagonal operator sending any real-rooted polynomial with all roots of the same sign to a real-rooted polynomial (not necessarily with roots of the same sign). There exist obvious versions of these notions for polynomials of bounded degree, e.g. a sequence γ = (γ0 , γ1 , ..., γk ) will be referred to as a multiplier sequence of length k + 1 or simply a finite multiplier sequence if it has the above mentioned properties when acting on the linear space Rk [x] of all polynomials of degree at most k. The following fundamental result was proved in [PS14]. Theorem 1. Let γ = {γj }, j = 0, 1, 2, . . . be a sequence of real numbers and Tγ : R[x] → R[x] be the corresponding diagonal operator. Define Φ(x) to be the formal power series ∞ X γ(k) k Φ(x) = x . k! k=0

The following assertions are equivalent: (i) γ is a multiplier sequence of the 1st kind; (ii) Φ(x) defines an entire function which is the limit, uniformly on compact sets, of polynomials with only real zeros of the same sign; Date: October 12, 2009. 2000 Mathematics Subject Classification. Primary 12D10, Secondary 32H99. Key words and phrases. multiplier sequence, discriminant amoeba. 1

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M. PASSARE, J. M. ROJAS, AND B. SHAPIRO

(iii) Either Φ(x) or Φ(−x) is an entire function that can be written as Φ(x) = Cxn eax Π∞ k=1 (1 + αk x), P∞ where n ∈ N, C ∈ R, a, αk ≥ 0 for all k ∈ N and k=1 αk < ∞; (iv) For all nonnegative integers n the polynomial Tγ [(1+x)n ] is real-rooted with all zeros of the same sign. Its finite-dimensional analog was proved 60 years later in [CC77], Theorem 3.7, see also [CC83], Theorem 3.1. Theorem 2. For γ = (γ0 , ..., γk ) the following two conditions are equivalent: (i) Tγ is a multiplier sequence of length k + 1 of the 1st kind; Pk (ii) The polynomial QTγ (t) = Tγ [(1 + x)k ] = j=0 kj γj tj has all real zeros of the same sign. Morally, these results are based on the log-concavity property of the space of all real-rooted polynomials. It turns out that, contrary to the classical case, there exists an interesting log-convex subset of all real-rooted polynomials which results in a completely different characterization of the set of its diagonal preservers. Definition 1. A real polynomial p(x) is called sign-independently real-rooted if it is real-rooted and all polynomials obtained by arbitrary sign changes of its coefficients are real-rooted as well. A simple example of a sign-independently real-rooted polynomial is xk + xk−1 . Less trivial examples can be found in § 2. Denote by SI ⊂ R[x] the set of all sign-independently real-rooted polynomials and denote by SI ≥0 ⊂ SI its subset consisting of polynomials with all non-negative coefficients. Analogously, denote by SIk ⊂ Rk the set of all sign-independently real-rooted polynomials of degree k and by SIk≥0 ⊂ SIk its subset consisting of polynomials with non-negative coefficients. An example of SI3≥0 is shown on Fig. 1. Finally, let SIk+ ⊂ SIk≥0 denote the set of all sign-independently real-rooted polynomials of degree k with strictly positive coefficients. The main result of this note is as follows. Theorem 3. Given a finite sequence γ = (γ0 , ..., γk ) of positive numbers one has that the diagonal operator Tγ : Rk [x] → Rk [x] preserves the set SIk+ if and only if for every j = 1, ..., k − 1 one has γj2 ≥ γj−1 γj+1 . Remark 1. The latter inequalities are usually called Tur´ an’s inequalities, see e.g. [CVV90], [CC04], Problem 4.8) and sequences of positive numbers satisfying these inequalities are called log-concave and they find frequent applications in combinatorics. Remark 2. Denote by G+ k the set of all log-concave positive sequences of length k+1 and denote by G∗k the set of all sequences γ = (γ0 , ..., γk ) with all non-vanishing entries which preserve SIk . Obviously, the set G+ k is a semigroup with respect to the usual Hadamard product which was shown already in [DaPo49]. Remark 3. Consider a linear map πk : Rk+1 → Rk [x]sending a finite sequence γ = (γ0 , ..., γk ) to the polynomial Pγ (x) = γ0 + nγ1 x + n2 x2 + ... + ni xi + ... + xn . Under this map the usual Hadamard product of sequences is mapped to the SchurSzeg˝ o product of polynomials, see e.g. [KSh06] and references therein.

ON A NEW TYPE OF MULTIPLIER SEQUENCES AND DISCRIMINANT AMOEBAE

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Figure 1. Three important domains for the family x3 + 3x2 + ax + b. Corollary 1. 1) The set G+ k is mapped by πk to the set of all polynomials of degree k with positive coefficients and satisfying Newton’s inequalities. In particular, all finite multiplier sequences of the first kind and length k + 1 belong to G+ k , see Theorem 2. 2) The semigroup G∗k of all finite sequence γ = (γ0 , ..., γk ) of non-vanishing numbers such that its diagonal operator Tγ : Rk [x] → Rk [x] preserves the set SIk is characterized by the inequalities γj2 ≥ |γj−1 γj+1 | for every j = 1, ..., k − 1. In k+1 particular, G∗k is obtained from G+ of all sign k by the action of the group Z2 changes. Remark 4. Recall that a real polynomial p(x) = a0 + a1 x + ... + ak xk satisfies New k 2 ton’s inequalities if Sj ≥ Sj−1 Sj+1 , j = 1, ..., k − 1 where Sj = aj / j . Sequences of positive numbers satisfying Newton’s inequalities are called ultra log-concave, see e.g. [Li97].

Remark 5. Using Descartes’ rule of signs one can show that if a sequence γ = (γ0 , ..., γk ) preserves SIk and, additionally, γ0 6= 0, γk 6= 0 then γ ∈ G∗k . Explanations to Fig. 1. There are three domains on the picture A ⊃ B ⊃ C each bounded by a pair of curved segments and an straight segment belonging to the vertical axis. Domain A is the domain of all real-rooted polynomials of the form x3 + 3x2 + ax + b with positive a and b satisfying Newton’s inequalities. Domain B is the domain of all polynomials of the above form and with all non-positive roots. Finally, domain C is the domain of all sign-independently real-rooted polynomials with non-negative coefficients. Acknowledgements. The second author thanks the Wenner Gren Foundation for the support of his visit to Stockholm University during the start of this project. Earlier related work of the second author was also partially supported by NSF CAREER grant DMS-0349309 and Sandia Laboratories. The authors are sincerely grateful to P. Br¨ anden for finding a mistake in the initial version of the paper and pointing out a number of relevant references.

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2. Proof To prove necessity of the Tur´an inequalities in Theorem 3 is very easy. One considers a sign-independently real-rooted polynomial xi−1 (1 + 2x + x2 ). Its image under the action of Tγ should be real-rooted which immediately gives γi2 ≥ γi−1 γi+1 . Proof of sufficiency in Theorem 3 (which we found) is more complicated and is based on the following construction. Consider the map Log| · | : (C∗ )k+1 → Rk+1 sending (C∗ )k+1 3 a 7−→ log |a0 |, log |a1 |, . . . , log |ak | ∈ Rk+1 , where a = (a0 , a1 , ..., ak ) ∈ (C∗ )k+1 . Notice that Log| · | maps Rk+1 differemorphi+ cally onto Rk+1 where R+ is the set of all positive real numbers. For any polynomial Q(a) ∈ C[a0 , ..., ak ] one defines its amoeba AQ as the image of the complex algebraic hypersurface HQ : {Q(a) = 0} under Log| · |. There is a natural 1 − 1-correspondence between unbounded connected components of the complement Rk+1 \ Af and the vertices of the Newton polytope of f . Namely, Lemma 1. [GKZ94, Prop. 1.7 & Cor. 1.8, pp. 195–196] Suppose a polynomial f ∈ C[x1 , . . . , xn ] has Newton polytope P and v is a vertex of P . Also let C denote the (open) cone of inner normals to v. Then there is a unique unbounded connected component of the complement to Af containing a translate of the cone C. In the above notation the corresponding translate of the cone C which sits inside a given unbounded connected component will be called its recession cone. For the class of the so-called A-discriminants more is known. (For the notion of A-discriminant consult [GKZ94, Ch. 9 & 12].) For example, for a A-discriminantal polynomial Q(a) the boundary of its amoeba AQ is contained in the image of the R R is of intersection of the real algebraic hypersurface HQ , where HQ real part HQ k+1 of HQ with R , see [GKZ94]. The latter fact motivates the following definition. For a real algebraic hypersurface HQ ⊂ Ck+1 given by Q(a0 , a1 , ..., ak ) = 0 † we define its complete reflection HQ as the union of 2k+1 hypersurfaces given by Q(±ao , ±a1 , ..., ±ak ) = 0 for all 2k+1 possible choices of signs of coordinates, see † R † Fig 2. Consider the restriction of real part (HQ ) of HQ to Rk+1 + . Notice that by † R k+1 the above remark each connected component of R+ \ (HQ ) is mapped by Log| · | diffeomorphically either onto a connected component of the complement Rk+1 \ AQ or onto AQ itself. One can say that AQ is the union of the images of some number of the latter connected components, see Fig. 2. Our main example of Q(a) is the discriminant of a univariate polynomial. Namely, consider a univariate polynomial P (t) = a0 + a1 t + a2 t2 + . . . + ak tk with variable coefficients. Its discriminant ∆ ∈ Z[a] is an irreducible polynomial in the variables a = (a0 , a1 , ..., ak ) vanishing precisely at those values of a for which P (t) has a multiple root. It is well known (see [GKZ94], p.271) that ∆ has the two homogeneity properties: ∆(λa0 , λa1 , λa2 , . . . , λak ) = λ2(k−1) ∆(a) and ∆(a0 , λa1 , λ2 a2 , . . . , λk ak ) = λk(k−1) ∆(a) . There is a combinatorial formula for those monomials in ∆ whose exponents correspond to the vertices of its Newton polytope (see [GKZ94], p.300 & p.302). Namely, each such vertex monomial corresponds to a unique subdivision of the line segment [0, k] into a collection of segments [0, k1 ][k1 , k2 ] . . . [km , k], with integers 0 < k1 < k2 < . . . < km < k. In particular, the finest subdivision [0, 1][1, 2] . . . [k − 1, k] of [0, k] into unit intervals is associated with the monomial


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Figure 2. Reflected discriminant for the family x3 + 3x2 + ax + b.

±a21 a22 · · · a2k−1 = ±(a1 a2 · · · ak−1 )2 ,

(1)

whereas the second finest subdivisions, having one segment [l − 1, l + 1] of length two and all other segments of unit length, correspond to the monomials 2 ±4 al−1 a−2 l al+1 (a1 a2 · · · ak−1 ) ,

l = 1, 2, . . . , k − 1 .

(2)

Combinatorially the Newton polytope of ∆ is a cube of dimension k − 1 and the monomials (2) represent its vertices v0 + el−1 − 2el + el+1 neighboring to the vertex v0 = (0, 2, 2, . . . , 2, 0) corresponding to the monomial (1). Denote by A∆ the amoeba of the discriminant ∆. Using the notation xl = log |al | we see that A∆ is the set of vectors (x0 , ..., xk ) ∈ Rk+1 such that the torus |a0 | = ex0 , . . . , |ak | = exk intersects the discriminant locus ∆(a) = 0. Denote by SIk+ the set of all sign-independently real-rooted polynomials with all simple and negative roots. Obviously, SIk≥0 is the closure of SIk+ . Proposition 1. The set SIk+ of all sign-independently real-rooted polynomials of degree k with all positive coefficients is mapped by Log| · | onto the connected component of the complement to A∆ represented by the monomial (1). Proof of this proposition is based on several additional statements. Let us first find more examples of sign-independently real-rooted polynomials. Consider the vector s ∈ Nk−1 given by k k k − j| + 1) + (| − j| + 2) + . . . + , 2 2 2 for k even, and by sj = (|

sj = (j −

j = 1, 2, . . . , k − 1,

k−1 k−1 k−1 ) + (j + 1 − ) + ... + , 2 2 2

j = 1, 2, . . . , k − 1,

for k odd. The first few instances of s are (1) for k = 2; (1, 1) for k = 3; (2, 3, 2) for k = 4; (2, 3, 3, 2) for k = 5; (3, 5, 6, 5, 3) for k = 6; (3, 5, 6, 6, 5, 3) for k = 7;

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(4, 7, 9, 10, 9, 7, 4) for k = 8; (4, 7, 9, 10, 10, 9, 7, 4) for k = 9; and (5, 9, 12, 14, 15, 14, 12, 9, 5) for k = 10. Lemma 2. The polynomial pk (x) = 1 + λs1 x + λs2 x2 + . . . + λsk−1 xk−1 + xk of degree k is sign-independently real-rooted for any sufficiently large value of the positive real parameter λ. Proof. This follows from the fact that for large λ the polynomial pk has coefficients approaching the polynomial qk given by: qk (x) = (x + λ−k/2 )(x + λ1−k/2 ) · · · (x + λk/2 ) if k is even, and by qk (x) = (x + λ−(k−1)/2 )(x + λ1−(k−1)/2 ) · · · (x + λ(k−1)/2 ) if k is odd. Indeed, in order to see that pk is real-rooted, one observes that the roots of qk are all real, and since they are given by distinct powers of λ, they are all of different magnitude. Hence, under the small change of real coefficients that is needed to deform qk to the original polynomial pk , the roots remain well apart, and hence cannot form any conjugate pair of complex roots. Now, one can easily check that for sufficiently large λ changing arbitrarily signs of roots of qk one obtains 2k polynomials close to 2k polynomials obtained from qk by arbitrary sign changes of its coefficients. Thus, any change of signs of some of the coefficients of pk just corresponds to an appropriate sign change in some of the roots of qk , and the preceding argument again shows that the polynomials are still real-rooted. ≥0 Lemma 3. The set SIk≥0 is fibered over SIk−1 with contractible 1-dimensional fibers.

Proof. Notice that the restriction of SIk≥0 to the hyperplane a0 = 0 is in obvious ≥0 1 − 1-correspondence with SIk−1 obtained by dividing a polynomial p(x) = a1 x + ... + ak xk from the former set by the variable x. To finish the proof we show that for any p(x) = a0 + a1 x + ... + ak xk belonging to SIk≥0 the family of polynomials pt (x) = p(x) − a0 τ, τ ∈ [0, 1] belong to SIk≥0 thus forming the required fiber of the projection in question. Indeed, consider for any real rooted polynomial P (x) = α0 + α1 x + ... + αk xk the family P (x) = P (x) + where ∈ R. It is obvious that P (x) is real-rooted if and only if ∈ [vmin , Vmax ] where vmin is the maximal local minimum of P (x) and Vmax is its minimal local maximum. Now take p(x) ∈ SIk≥0 and consider its family p (x). Assuming that p(x) = a0 +a1 x+...+ak xk with all ai ≥ 0 consider p− (x) = −a0 + a1 x + ... + ak xk which is also real-rooted. Thus at least for in the interval [−2a0 , 0] one has that p (x) is real-rooted. Exactly the same argument works for all p± (x) obtained from p(x) by arbitrary sign changes of its coefficients proving that the family p(x) − a0 τ, τ ∈ [0, 1] sits inside SIk≥0 . Denote by Ck the connected component of Rk+1 \A∆ corresponding to the monomial (1). Lemma 4. The set G+ k of all finite sequences γ = (γ0 , ..., γk ) of positive numbers whose corresponding diagonal operator Tγ preserves SIk+ is isomorphically mapped by Log| · | onto the recession cone of Ck ⊂ Rk+1 . Proof. Indeed, multiplication by a diagonal sequence (γ0 , ..., γk ), γi > 0 in coordinates (a0 , ..., ak ) corresponds to addition of the vector (log γ0 , ..., log γk ) in coordinates (x0 , ..., xk ). Since Ck = Log(SIk+ ) ⊂ Rk+1 is convex it is preserved under an


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affine shift by some vector if and only if this vector lies in the minimal convex cone containing it, i.e. in its recession cone. Denoting by Kk the recession cone of Ck we now show the following crucial statement. Lemma 5. The cone Kk is given by the inequalities 2xl ≥ xl−1 + xl+1 , for l = 1, 2, . . . , k − 1. Proof. Recall that for a polynomial P (z) in n complex variables z = (z1 , ..., zn ), one defines its Ronkin function by the formula Z dz1 dz2 1 dzn log |P (z)| NP (x) = ∧ ∧ ... ∧ , (2πi)n Log−1 (x) z1 z2 zn where x = (x1 , x2 , ..., xn ). Equivalently, NP is given by the integral Z 1 log |P (z)|dθ1 dθ2 ...dθn , NP (x) = (2π)n [0,2π]n where z = ex1 +iθ1 , ex2 +iθ2 , ..., exn +iθn . It is known that the Ronkin function is convex, and it is affine on each connected component of the complement of the amoeba AP of P (z). As an example, the Ronkin function of a monomial P (z) = az1l1 z2l2 ...znln , a 6= 0 is given by NP (x) = log |a| + l1 x1 + l2 x2 + ... + ln xn . It was shown in [PT04] that the vertex monomials of the univariate discriminant ∆ are in 1 − 1-correspondence with the connected components of the complement of the amoeba A∆ . Additionally, from general results proved in [PR04] one knows that the Ronkin function of ∆ is equal to log |cv |+hv, xi in the component corresponding to a vertex monomial cv xv . In particular, in the components of the special vertex monomials (1) and (2), the Ronkin function coincides with the affine linear functions 2x1 + 2x2 + . . . + 2xk−1 = 2 (x1 + x2 + . . . + xl−1 ) and +xl−1 − 2xl + xl+1 + 2 (x1 + x2 + . . . + xk−1 ) respectively. Now, by [PST05] one knows that the discriminant amoeba does not have any other unbounded components of the complement to the amoeba except the ones corresponding to its vertices. Therefore, it follows that its spine S∆ (see details in [PR04]) is defined explicitly as the corner locus of the piecewise linear convex function (or tropical polynomial) max log |cv | + hv, xi , v

where v ranges over all the vertices of the Newton polytope of ∆. The connected components of the complement of the spine S∆ are convex polyhedral cones where one of the affine linear functions dominates all the others, and the closure of such a cone is the recession cone of the unbounded connected component of the complement to A∆ . For the special vertex monomial (1) we obtain in this way that the recession cone Kk of Ck is given by the inequalities: 2 (x1 +x2 +. . .+xk−1 ) ≥ xl−1 −2xl +xl+1 +2 (x1 +x2 +. . .+xk−1 ), l = 1, 2, . . . , k−1 , or, equivalently, 2xl ≥ xl−1 + xl+1 , for l = 1, 2, . . . , k − 1.

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Proof. [Proof of Proposition 1] Obviously, the set SIk+ (if non-empty) consists of some number of connected components of the complement Rk+1 \ ∆† where ∆† is the reflected discriminant of ∆, see Fig. 2. Indeed, SI + is the intersection of the set of all real-rooted polynomials with simple zeros with all similar sets obtained by all possible sign changes of the coefficients. By Lemmas 1-2 the set SIk+ is non-empty and connected thus coinciding with a unique connected component of Rk+1 \ ∆† . To finish the proof of Proposition 1 we have to show that the image of SIk+ under Log| · | coincides with the component of the complement to A∆ corresponding to the monomial (1). We show that the vector s ∈ Nk−1 from Lemma 1 is an interior point in the recession cone of the special (2, 2, . . . , 2)-component of the complement of the discriminant amoeba. Indeed, this recession cone is defined by the inequalities 2xj ≥ xj−1 + xj+1 , j = 1, 2, . . . , k − 1 with the dehomogenizing convention x0 = xk = 0. This means that the coefficients λsj of the polynomial pk from Lemma 1, for large enough λ, represent a point in the special component of the amoeba complement. But the polynomial pk was seen to be sign-independently real-rooted for large λ, and this concludes the proof. Taking exponential functions of the inequalities describing Kk we finish the proof of Theorem 3. Finally, Part 1) of Corollary 1 is immediate from the definition of πk and the fact that every real polynomial with all real negative roots satisfies Newton’s inequalities. Part 2) follows immediately from the invariance of SIk under the group Zk+1 of all sign changes of the coefficients of polynomials of degree (at most) k. 2 3. Final remarks Problem 1. How to count connected components of the complement to the reflected discriminant of a given discriminant? In particular, is it true that the number of connected components of the complement to the reflected discriminant of univariate polynomials of degree k restricted to Rk+ equals 2k , see Fig. 2? Problem 2. Find an elementary proof of Theorem 3 avoiding the usage of discriminant amoebae. Problem 3. Describe the set of all diagonal sequences preserving the set of all signindependently non-negative polynomials, i.e. those which are non-negative and stay non-negative under the action of the above group Zk+1 of sign changes. 2 References [CVV90] G. Csordas, R. Varga, and I. Vincze, Jensen polynomials with applications to the Riemann ζ-function, JMAA 153(1) (1990), 112–135. [CC83] T. Craven and G. Csordas, Location of zeros. I. Real polynomials and entire functions. Illinois J. Math. 27(2) (1983), 244–278. [CC77] T. Craven and G. Csordas, Multiplier sequences for fields. Illinois J. Math. 21(4) (1977), 801–817. [CC04] T. Craven, and G. Csordas, Composition theorems, multiplier sequences and complex zero decreasing sequences. in Value distribution theory and related topics, 131–166, Adv. Complex Anal. Appl., 3, Kluwer Acad. Publ., Boston, MA, 2004. [DaPo49] H. Davenport, G. Polya, On the product of two power series, Cand. J. Math. 1, (1949), 1–5. [GKZ94] I. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Reprint of the 1994 edition. Modern Birkhuser Classics. Birkh¨ auser Boston, Inc., Boston, MA, 2008. x+523 pp. [KSh06] V. Kostov, and B. Shapiro, On the Schur-Szeg¨ o composition of polynomials, C. R. Math. Acad. Sci. Paris, 343(2) (2006), 81–86. [Li97] T. Liggett, Ultra logconcave sequences and negative dependence. J. Combin. Theory Ser. A 79(2) (1997), 315–325.


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¨ [PS14] G. P´ olya and J. Schur, Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113. [PR04] M. Passare, H. Rullg˚ ard, Amoebas, Monge–Amp` ere measures and triangulations of the Newton polytope, Duke Math. J., 121 (2004), 481–507. [PST05] M. Passare, T. Sadykov, A. Tsikh, Singularities of hypergeometric functions in several variables, Compos. Math., 141 (2005),787–810. [PT04] M. Passare, A. Tsikh, Algebraic equations and hypergeometric series, pp. 653–672 in ”The legacy of Niels Henrik Abel”, Springer, Berlin, 2004. [Sur92] A survey of linear preserver problems. Linear and Multilinear Algebra, 33 (1992), no. 1-2, pp. 1–129. Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden E-mail address: [email protected] Department of Mat hematics, Texas A&M University, 3368 TAMU, College Station, Texas 77843, USA E-mail address: [email protected] Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden E-mail address: [email protected]