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1 SIAM J. SCI. COMPUT. Vol. 38, No. 2, pp. A765 A788 c 206 Societ for Industrial and Applied Mathematics ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS VIA DETERMINANTAL REPRESENTATIONS BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH Abstract. We give two determinantal representations for a bivariate polnomial. The ma be used to compute the zeros of a sstem of two of these polnomials via the eigenvalues of a twoparameter eigenvalue problem. The first determinantal representation is suitable for polnomials with scalar or matri coefficients and consists of matrices with asmptotic order n 2 /4, where n is the degree of the polnomial. The second representation is useful for scalar polnomials and has asmptotic order n 2 /6. The resulting method to compute the roots of a sstem of two bivariate polnomials is ver competitive with some eisting methods for polnomials up to degree 0, as well as for polnomials with a small number of terms. Ke words. sstem of bivariate polnomial equations, determinantal representation, twoparameter eigenvalue problem, polnomial two-parameter eigenvalue problem AMS subject classifications. 65F5, 65H04, 65F50, 3P5 DOI. 0.37/ Introduction. In this paper, we make progress on a problem that has essentiall been open since Dion s 902 paper [0]. It is well known that for each monic polnomial p() = p 0 + p + + p n n + n one can construct a matri A C n n such that det(i A) = p(). One of the options is a companion matri (see, e.g., [2, p. 46]) A p = p 0 p p n Thus, we can numericall compute the zeros of the polnomial p as eigenvalues of the corresponding companion matri A p using tools from numerical linear algebra. This approach is used in man numerical packages, for instance, in the roots command in MATLAB [30].. Submitted to the journal s Methods and Algorithms for Scientific Computing section August 26, 204; accepted for publication (in revised form) December 28, 205; published electronicall March 5, Department of Mathematics, Universit of Ljubljana, Jadranska 9, SI-000 Ljubljana, Slovenia (bor.plestenjak@fmf.uni-lj.si). The research was performed while this author was visiting the CASA group at TU Eindhoven. Department of Mathematics and Computer Science, TU Eindhoven, PO Bo 53, 5600 MB, The Netherlands ( hochsten/). This author s work was supported b an NWO Vidi grant. A765

2 A766 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH The aim of this paper is to find a similar elegant tool for finding the zeros of a sstem of two bivariate polnomials of degree n, (.) p(, ) := q(, ) := n i=0 n i=0 n j j=0 n j j=0 p ij i j = 0, q ij i j = 0. An approach analogous to the univariate case would be to construct matrices A, B, C, A 2, B 2, and C 2 of size n n such that (.2) det(a + B + C ) = p(, ), det(a 2 + B 2 + C 2 ) = q(, ). This would give an equivalent two-parameter eigenvalue problem [] (.3) (A + B + C ) u = 0, (A 2 + B 2 + C 2 ) u 2 = 0 that could be solved b the standard tools like the QZ algorithm; see [7] for details. This idea looks promising, but there are man obstacles on the wa to a working numerical algorithm that could be applied to a sstem of bivariate polnomials. Although it has been known for more than a centur [9, 0, 6] that such matrices of size n n eist, so far there are no efficient numerical algorithms that can construct them. Even worse, it seems that the construction of such matrices might be an even harder problem than finding zeros of polnomials p and q. There eist simple and fast constructions [32, 40] that build matrices of size O(n 2 ) that satisf (.2), where the resulting two-parameter eigenvalue problem (.3) is singular; we will discuss more details in section 4. Recent results [32] show that it is possible to solve singular two-parameter eigenvalue problems numericall for small to medium-sized matrices. However, the O(n 2 ) size of the matrices pushes the compleit of the algorithm to the enormous O(n 2 ) and it is reported in [3] that this approach to compute zeros is competitive onl for polnomials of degree n < 5. The construction of [32] ields matrices that are of asmptotic order 2 n2, while those of [40] are of asmptotic order 4 n2. In this paper we give two new representations. The first one uses the tree structure of monomials in and. The resulting matrices are smaller than those of [40], with the same asmptotic order 4 n2. This representation can be used for bivariate polnomials as well as for polnomial twoparameter eigenvalue problems [33], that is, for polnomials with matri coefficients. The second representation is even more condensed, with asmptotic order 6 n2, and can be applied to scalar bivariate polnomials. Although the size of the matrices asmptoticall still grows quadraticall with n, the smaller size renders this approach attractive for polnomials of degree n < 0, or for larger n if the polnomials have onl a few terms. This alread is an interesting size for a practical use and might trigger additional interest in such methods that could culminate in even more efficient representations. Moreover, as we will see, for modest n, the order of the matrices is onl roughl 2n. Furthermore, for polnomials of degree 3, we present a new construction of matrices of order (eactl) 3.

3 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A767 There are other was to stud a sstem of polnomials as an eigenvalue problem (see, e.g., [, 44]), but the involve more smbolic computation. In [28], an algorithm is proposed that onl requires to solve linear sstems and check rank conditions, which are similar tools that we use in the staircase method [32] to solve the obtained singular two-parameter eigenvalue problem. Of course, there are man numerical methods that can be applied to sstems of bivariate polnomials. Two main approaches are the homotop continuation and the resultant method; see, e.g., [2, 23, 4, 45, 49] and the references therein. There are also man methods which aim to compute onl real solutions of a sstem of two real bivariate polnomials; see, e.g., [34, 43]. We compare our method with several eisting approaches, including Mathematica s NSolve [5] and PHCpack [49], in section 7 and show that our approach is ver competitive for polnomials up to degree < 0. We mention that another advantage of writing the sstem of bivariate polnomials as a two-parameter eigenvalue problem is that then we can appl iterative subspace numerical methods such as the Jacobi Davidson method and compute just a small part of zeros close to a given target ( 0, 0 ) [9]; we will not pursue this approach in this paper. The rest of this paper is organized as follows. In section 2 we give some applications where bivariate polnomial sstems have to be solved. Section 3 introduces determinantal representations, and section 4 focuses on two-parameter eigenvalue problems. In section 5 we give a determinantal representation that is based on the tree of monomials, involves no computation, and is suitable for both scalar and matri polnomials. The matrices of the resulting representation are asmptoticall of order 4 n2. In section 6 we give a representation with smaller matrices, of asmptotic order 6 n2, that involves just a trivial amount of numerical computation (such as computing roots of low-degree univariate polnomials) and can be computed ver efficientl. This representation ma be used for scalar polnomials. We end with some numerical eperiments in section 7 and conclusions in section Motivation. Polnomial sstems of form (.) arise in numerous applications and fields, such as signal processing [5, 8, 4, 47] and robotics [53]. In computer aided design, one ma be interested in the intersections of algebraic curves, such as ellipses [2, 26, 29]. In two-dimensional subspace minimization [7], such as polnomial tensor optimization, one is interested in two-dimensional searches min α,β F ( + αd + βd 2 ), where F : R n R, is the current point, and d and d 2 are search directions; see [42, 43] and the references therein. In dela differential equations, determining critical delas in the case of so-called commensurate delas ma lead to a problem of tpe (.) [22]. The simplest eample is of the form (t) = a (t) + b (t τ) + c (t 2τ), where τ > 0 is the dela; asked are values of τ that result in periodic solutions. This ields p and q of degrees 2 and 3, respectivel. Taking more dela terms with delas that are multiples of τ gives polnomials of higher degree. In sstems and control the first-order conditions of the L 2 -approimation problem of minimizing h h 2 = 0 h(t) h(t) 2 dt, for a given impulse response h of degree n, and degree( h) = ñ n, lead to a sstem of tpe (.) [3]. When considering quadratic eigenvalue problems in numerical linear algebra, it is of interest to determine argmin θ C (θ 2 A + θb + C)u as an approimate eigenvalue for a given approimate eigenvector u, which gives a sstem of degree 3 in the real and imaginar part of θ [20, section 2.3]. Generalizations to polnomial eigenvalue problems give rise to polnomials p and q of higher degree.

4 A768 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH Also, there has been some recent interest in this problem in the contet of the chebfun2 project [34, 46]. In chebfun2, nonlinear real bivariate functions are approimated b bivariate polnomials, so solving (.) is relevant for finding zeros of sstems of real nonlinear bivariate functions and for finding local etrema of such functions. 3. Determinantal representations. In this section we introduce determinantal representations and present some eisting constructions. The difference between what should theoreticall be possible and what can be done in practice is huge. The algorithms we propose reduce the difference onl b a small (but still significant) factor; there seems to be plent of room for future improvements. We sa that a bivariate polnomial p(, ) has degree n if all its monomials p ij i j have total degree less than or equal to n, i.e., i + j n, and if at least one of the monomials has total degree equal to n. We sa that the square m m matrices A, B, and C form a determinantal representation of the polnomial p if det(a + B + C) = p(, ). As our motivation is to use eigenvalue methods to solve polnomial sstems, we will, instead of determinantal representation, often use the term linearization since a determinantal representation transforms an eigenvalue problem that involves polnomials of degree n into a linear eigenvalue problem (.3). A definition of linearization that etends that for the univariate case (see, e.g., [27]) is the following. Definition 3.. A linear bivariate pencil A + B + C of size m m is a linearization of the polnomial p(, ) if there eist two polnomial matrices L(, ) and Q(, ) such that det(l(, )) det(q(, )) and L(, ) (A + B + C) Q(, ) = [ ] p(, ) 0. 0 I m We are interested not onl in linearizations of scalar polnomials but also in linearizations of matri bivariate polnomials of the form (cf. (.)) (3.) P (, ) = n i=0 n j j=0 i j P ij, where the P ij are k k matrices. In line with the above, a linear pencil A+B +C of matrices of size m m presents a linearization (determinantal representation) of the matri polnomial P (, ) if there eist two polnomial matrices L(, ) and Q(, ) such that det(l(, )) det(q(, )) and L(, ) (A + B + C) Q(, ) = [ ] P (, ) 0. 0 I m k In this case det(a + B + C) = det(p (, )). Each linearization of a matri polnomial gives a linearization for a scalar polnomial, as we can think of scalars as of matrices; the opposite is not true in general. Dion [0] showed that for ever scalar bivariate polnomial p(, ) of degree n there eists a determinantal representation with smmetric matrices of size n n. Dickson [9] later showed that this result cannot be etended to general polnomials in more than two variables, ecept for three variables and polnomials of degree two and three, and four variables and polnomials of degree two. Although the both give constructive proofs, there does not seem to eist an efficient numerical algorithm

5 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A769 to construct the determinantal representation with matrices of size n n for a given bivariate polnomial of degree n. In recent ears, the research in determinantal representations has been growing, as determinantal representations for a particular subset of polnomials, real zero polnomials, are related to linear matri inequalit (LMI) constraints used in semidefinite programming (SDP). For an overview see, e.g., [36, 50]; here we give just the essentials for bivariate polnomials that are related to our problem. We sa that a real polnomial p(, ) satisfies the real zero condition with respect to ( 0, 0 ) R 2 if for all (, ) R 2 the univariate polnomial p (,) (t) = p( 0 + t, 0 + t) has onl real zeros. A two-dimensional LMI set is defined as { (, ) R 2 : A + B + C 0 }, where A, B, and C are smmetric matrices of size m m and 0 stands for positive semidefinite. In SDP we are interested in conve sets S R 2 that admit an LMI representation, i.e., S is an LMI set for certain matrices A, B, and C. Such sets are called spectrahedra and Helton and Vinnikov [6] showed that such S must be an algebraic interior, whose minimal defining polnomial p satisfies the real zero condition with respect to an point in the interior of S. Their results state that if a polnomial p(, ) of degree n satisfies real zero condition with respect to ( 0, 0 ), then there eist smmetric matrices A, B, and C of size n n such that det(a + B + C) = p(, ) and A + 0 B + 0 C 0. Matrices A, B, and C thus form a particular determinantal representation for p. The problem of constructing an LMI representation with smmetric or Hermitian matrices A, B, and C for a given spectrahedron S raised much more interest than the related problem of generating a determinantal representation for a generic bivariate polnomial. There eist procedures, which rel heavil on slow smbolic computation or other epensive steps, that return an LMI representation with Hermitian matrices for a given spectrahedron, but the are not efficient enough. For instance, a method from [38], based on the proof from [0], does return n n matrices for a polnomial of degree n, but the reported times (0 seconds for a polnomial of degree 0) show that it is much too slow for our purpose. As a first step of the above method is to find zeros of a sstem of bivariate polnomials of degree n and n, this clearl cannot be efficient enough for our needs. In addition, we are interested in determinantal representations for polnomials that do not necessar satisf the real zero condition. In SDP and LMI the matrices have to be smmetric or Hermitian, which is not required in our case. We need a simple and fast numerical construction of matrices that satisf (.2) and are as small as possible ideall their size should increase linearl and not quadraticall with n. Regarding available determinantal representations for generic bivariate polnomials, we first have the linearization b Khazanov with matrices of size n 2 n 2 [25]. In [33, Appendi], a smaller linearization for bivariate matri polnomials is given with block matrices of order 2n(n + ). The linearization uses all monomials of degree up to n and contains a direct epression for the matrices A, B, and C such that det(a + B + C) = p(, ). Similar to [25], it can be applied to matri polnomials. We give an eample for a general matri polnomial of degree 3, from which it is possible to deduce the construction for a generic degree. This linearization will be superseded in section 5 b a more economical one.

6 A770 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH Eample 3.2 (see [33, Appendi]). We take a matri bivariate polnomial of degree 3, P (, ) = P 00 +P 0 +P P 20 +P + 2 P P P P P 03. If u is a nonzero vector, then P (, )u = 0 if and onl if (A + B + C)u = 0, where (3.2) P 00 P 0 P 0 P 20 + P 30 P + P 2 P 02 + P 2 + P 03 I k I k I A + B + C = k 0 I k I k 0 I k I k 0 I k I k 0 0 I k and u = u [ 2 2 ] T. We have det(a + B + C) = det(p (, )) and A + B + C is a linearization of P (, ). We remark that Quarez [40] also gives eplicit epressions for determinantal representations. He is interested in smmetric representations and is able to construct, for a bivariate polnomial of degree n such that p(0, 0) 0, a linearization with smmetric matrices of size N N, where ( ) n/2 + 2 (3.3) N = 2 n This has asmptoticall the same order as the first linearization that we will give in section 5; the second linearization in section 6 has a smaller order. We also remark that in the phase, when we are solving a two-parameter eigenvalue problem to compute the zeros of a sstem of two bivariate polnomials, we cannot eploit the fact that the matrices are smmetric, so this is not important for our application. There are some other available tools, for instance, it is possible to construct a determinantal representation using the package NCAlgebra for noncommutative algebra [5, 35] that runs in Mathematica [5], but this does not give satisfactor results for our application as the matrices that we can construct have smaller size. 4. Two-parameter eigenvalue problems. In this section we briefl present the two-parameter eigenvalue problem and some available numerical methods. A motivation for the search for small determinantal representations is that if we transform a sstem of bivariate polnomials into an eigenvalue problem, then we can appl eisting numerical methods for such problems. A two-parameter eigenvalue problem has the form (.3), where A i, B i, and C i are given n i n i comple matrices. We are looking for, C and nonzero vectors u i C ni, i =, 2, such that (.3) is satisfied. In such case we sa that a pair (, ) is an eigenvalue and the tensor product u u 2 is the corresponding eigenvector. If we introduce the so-called operator determinants, the matrices (4.) 0 = B C 2 C B 2, = C A 2 A C 2, 2 = A B 2 B A 2,

7 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A77 then the problem (.3) is related to a coupled pair of generalized eigenvalue problems (4.2) w = 0 w, 2 w = 0 w for a decomposable tensor w = u u 2. If 0 is nonsingular, then Atkinson [] showed that the solutions of (.3) and (4.2) agree and the matrices 0 and 0 2 commute. In the nonsingular case the two-parameter problem (.3) has n n 2 eigenvalues and we can numericall solve it with a variant of the QZ algorithm on (4.2) from [7]. Ideall, if we could construct a determinantal representation with matrices n n for a bivariate polnomial of degree n, this would be the method that we would appl on the companion two-parameter eigenvalue problem to get the zeros of the polnomial sstem. As 0,, and 2 have size n n 2 n n 2, the computation of all eigenvalues of a nonsingular two-parameter eigenvalue problem has time compleit O(n 3 n 3 2), which would lead to O(n 6 ) algorithm for a sstem of bivariate polnomials. Of course, for this approach we need a construction of a determinantal representation with matrices n n that should not be more computationall epensive than the effort to solve a two-parameter eigenvalue problem. Unfortunatel, all practical constructions for determinantal representations (including the two presented in this paper) return matrices that are much larger than n n. If we have a determinantal representation with matrices larger than the degree of the polnomial, then the corresponding two-parameter eigenvalue problem is singular, which means that both matri pencils (4.2) are singular, and we are dealing with a more difficult problem. There eists a numerical method from [33] that computes the regular eigenvalues of (.3) from the common regular part of (4.2). For the generic singular case it is shown in [32] that the regular eigenvalues of (.3) and (4.2) agree. For other tpes of singular two-parameter eigenvalue problems the relation between the regular eigenvalues of (.3) and (4.2) is not completel known, but the numerical eamples indicate that the method from [33] can be successfull applied to such problems as well. However, the numerical method, which is a variant of a staircase algorithm [48], has to make a lot of decisions on the numerical rank and a single inaccurate decision can cause the method to fail. As the size of the matrices increases, the gaps between singular values ma numericall disappear and it ma be difficult to solve the problem. This is not the onl issue that prevents the use of determinantal representations to solve a bivariate sstem. The algorithm for the singular two-parameter eigenvalue problems still has compleit O(n 3 n 3 2), but the fast determinantal representations that we are aware of return matrices of size O(n 2 ) instead of O(n). This is what pushes the overall compleit to O(n 2 ) and makes this approach efficient onl for polnomials of small degree. Nonetheless, at compleit so high, each construction that gives a smaller determinantal representation can make a change. In view of this, we propose two new linearizations in the net two sections. 5. First linearization. We are interested in linearizations of the matri polnomial P (, ) = P 00 + P 0 + P n P n0 + n P n, + + n P 0n of degree n, where P ij are square matrices. Our goal is to find square matrices A, B, and C as small as possible such that det(a + B + C) = det(p (, )). Also, we need a relation that P (, )u = 0 if and onl if (A + B + C)u = 0, where u is a

8 A772 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH tensor product of u and a polnomial of and. The linearization in this section also applies to scalar bivariate polnomials, where all matrices are and u =. In section 3 we have given a linearization with block matrices of order 2n(n + ). We can view this linearization in the following wa. If P (, )u = 0 for u 0, then det(a + B + C)u = 0, where the vector u has the form (5.) u = u [ 2 2 n n 2 n ] T. This means that u alwas begins with the initial block u and then contains all blocks of the form j k u, where j + k n. To simplif the presentation we will usuall omit u when referring to the blocks of the vector (5.). The blocks are ordered in the degree negative leicographic ordering, i.e., a b c d if a+b < c+d, or a+b = c+d and a > c. The above block structure of vector u is defined in the rows of the matri from the second one to the last one (see Eample 3.2). For each block s = j k of (5.) such that j + k there alwas eists a preceding block q of the grade j + k such that either s = q or s = q (when j and k both options are possible). Suppose that s = q, ind(s) = i s, and ind(q) = i q, where function ind returns the inde of a block. Then the matri A + B + C has block I on position (i s, i q ) and block I on position (i s, i s ). These are the onl nonzero blocks in the block row i s. A similar construction with I replaced b I is used in the case s = q. The first block row of the matri A + B + C is used to represent the matri polnomial P (, ). One can see that there eist linear pencils A i + B i + C i, i =,..., m, such that (5.2) P (, )u = [ A + B + C A m + B m + C m ] u, where m = 2 n(n + ) is the number of blocks in (5.). The pencils in (5.2) are not unique. For instance, a term j k P jk of P (, ) can be represented in one of up to three possible was: (a) if j + k < n, we can set A p = P jk, where p = ind( j k ), (b) if j > 0, we can set B p = P jk, where p = ind( j k ), (c) if k > 0, we can set C p = P jk, where p = ind( j k ). Based on the above discussion we see that not all the blocks in (5.) are needed to represent a matri polnomial P (, ). What we need is a minimal set of monomials j k, where j + k < n, that is sufficient for a matri polnomial of degree n. We can formulate the problem of finding the smallest possible set for a given polnomial as a graph problem. We can think about all possible terms j k, where j + k < n, as of nodes in a directed graph G with the root and a directed edge from node s to node t if t = s or t = s (see Figure for the case n = 5). Now, we are looking for the smallest connected subgraph G with a root that can represent a given polnomial. Equivalentl, we are looking for a minimal directed rooted tree. Let us remember that for each term j k P jk of the polnomial P (, ) there are up to three possible nodes in the graph G that can be used to represent it. It is sufficient that one of these nodes is in a minimal tree G. Furthermore, if j + k >, then we can assume that we alwas use a node of degree j + k to represent j k P jk and then there are onl one or two options for a given term. All together, each nonzero term j k P jk, where j + k > 0, in the polnomial P defines one of the following rules for the subgraph G : (a) if k = 0, then j 0 has to be in the subgraph G, (b) if j = 0, then 0 k has to be in the subgraph G,

9 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A Fig.. Graph G for the polnomial of degree 5. (c) if j > 0, and k > 0, then at least one of j k or j k has to be in the subgraph G. The term P 00 can be presented b the root, which is alwas present in the subgraph G. Finding a minimal tree for a given polnomial is not an eas problem: it can be formulated as an NP-hard directed Steiner tree problem (DST) (see, e.g., [24]), where one has a directed graph G = (V, E) with nonnegative weights on edges and the goal is to find the minimum weight directed rooted tree that connects all terminals X V to a given root r V. Suppose that we are looking for a minimal representation tree for a polnomial P (, ) of degree n. In the graph G, which contains all nodes j k for j + k < n (see Figure for the case n = 5), we put weight on all directed and edges. Now we add a new verte for each monomial j k that is present in P (, ) and connect it with zero weight edges from all possible nodes in G that could be used to represent the monomial in the linearization. We make a DST problem b taking node as a root and all newl added vertices as terminals. From a solution of the DST problem the minimal representation tree can be recovered. Although this is an NPhard problem, there eist some polnomial time approimation algorithms that give a solution close to the optimal one and could be used to construct a small determinantal representation for a given polnomial with small number of nonzero terms. For the latest available algorithms, see, e.g., [4] and the references therein. Eample 5.. We are interested in a minimal tree for the matri polnomial (5.3) P (, ) = P 00 + P 0 + P P P P P P P 24. Nonzero terms in (5.3) define the nodes that have to be present in the minimal subgraph. The either are strictl defined as are the nodes, 2, and 5 or come in pairs where at least one element of each pair has to be present in the subgraph. Such pairs are ( 2, 2 ), ( 4, 3 ), and ( 2 3, 4 ). The situation is presented in Figure 2, where nodes and pairs, such that either the left or the right node has to be included, are shadowed in green. The nodes of the minimal connected subgraph that includes all required nodes are colored red. In a DST formulation each green shadow presents a terminal linked b zero weight edges to one or two nodes that are included in the region. On all other edges we put weight and then search for the minimum weight directed rooted tree that connects all terminals to the root.

10 A774 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH Fig. 2. A minimal directed subtree G for the matri polnomial (5.3) of degree 6. Matri polnomial (5.3) can thus be represented with matrices of block size. If we order the nodes of the subgraph in the degree negative leicographic ordering, then u has the form u = u [ ] T and a possible first block row of A + B + C has the form [ P0 + P 0 + P P 03 0 P 22 P 4 P 4 P 60 P 24 ]. In the subsequent block rows, the matri A + B + C has onl 20 nonzero blocks, 0 of them identit blocks on the main diagonal. The remaining nonzero blocks are I on block positions (2, ), (4, 2), (6, 4), (7, 5), (8, 6), (0, 8), (, 9) and blocks I on positions (3, ), (5, 3), (9, 7). If we have a generic matri polnomial P (, ), whose terms are all nonzero, then it is eas to see that the subgraph that contains all terms j k, where j + k < n and either j = 0 or k is even, is minimal. The detailed situation for the case n = 6 is presented in Figure 3, and representation trees for polnomials of degree to 8 are presented in Figure 4. Counting the number of nodes in the tree gives the following result: { (5.4) ψ(n) := G = 4 4 n(n + ), n even, (n )(n + 5) +, n odd. If we compare this with the linearization from Eample 3.2 that has matrices of block size 2n(n + ), we see that the new linearization uses matrices of roughl half size. The size of the matrices is also smaller than (3.3) from [40], while having the same asmptotic order. Theorem 5.2. We can linearize each matri polnomial P (, ) of degree n with matrices of block size ψ(n) from (5.4) using a minimal tree G that contains the terms j k, where j + k < n and either j = 0 or k is even.

11 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A Fig. 3. A minimal tree G for a generic polnomial of degree 5. ψ() = ψ(2) = 3 ψ(3) = 5 ψ(4) = 8 ψ(5) = ψ(6) = 5 ψ(7) = 9 ψ(8) = 24 Fig. 4. Minimal representation trees for polnomials of degrees to 8. Proof. We order all nodes of a minimal tree G in the degree negative leicographic ordering and form the block matri L(, ) in the following wa. All diagonal blocks of L(, ) are I. If a node with inde p is connected to a node with inde q with an or edge, then we put I or I in the block position (q, p), respectivel. Because of the ordering, the matri L is block lower triangular and nonsingular. Its inverse L(, ) is therefore also a lower triangular matri with diagonal identit blocks. Let m = ψ(n) be the number of nodes in G. If follows from L(, )L(, ) = I that the first block column of L(, ) has the form (5.5) I [ s 2 s 3 s m ] T, where s j is the monomial in the jth node of G for j =,..., m (s = ). Now we will construct the linearization of the matri polnomial P (, ). We need a block matri M(, ) = A + B + C, whose elements are linear pencils in and. We take M(, ) = L(, ) and adjust the first block row M (, ), where we put linear pencils such that M (, )(I [ s 2 s 3 s m ] T ) = P (, ). This is alwas possible as for each term j k P jk in the polnomial P (, ) there eists a term r q in G such that (j, k) (r, q) is one of the following three options: (0, 0), (, 0), or (0, ). The product M(, )L(, ) is an upper block triangular matri of

12 A776 the form BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH P (, ) H 2 (, ) H m (, ) M(, )L(, ) I =..., I where H 2 (, ),..., H m (, ) are matri polnomials. If we introduce the matri polnomial I H 2 (, ) H m (, ) I U(, ) =..., I then it follows that P (, ) U(, ) M(, ) L(, ) = I..., I and since det(l(, )) det(u(, )), this proves that M(, ) = A + B + C is indeed a linearization of the matri polnomial P (, ). Eample 5.3. As an eample we consider the scalar bivariate polnomial p(, ) = , which was alread linearized in [32] with matrices of size 6 6 (we can also get a 6 6 linearization if we insert the coefficients in matri (3.2) of Eample 3.2). Now we can linearize it with matrices of size 5 5 as p(, ) = det(a + B + C), where A + B + C = In the net section we will further reduce the size of the matrices to 4 4 and even Second linearization. We will upgrade the approach from the previous section and produce even smaller representations for scalar polnomials. As before, representations have a form of the directed tree, but instead of using onl and, an edge can now be an linear polnomial α + β such that (α, β) (0, 0). These additional parameters give us enough freedom to produce smaller representations. The root is still, while the other nodes are polnomials in and that are products of all edges on the path from the root to the node. In each node all monomials have the same degree, which is equal to the graph distance to the root. Before we continue with the construction, we give a small eample to clarif the idea.. Eample 6.. A linearization of a polnomial of degree 3 with matrices of size 4 4 is presented in Figure 5. We now eplain the figure and show how to produce

13 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A q 2 q 4 2 q 3 Fig. 5. A representation tree and a linearization for a polnomial of degree 3. the matrices from the representation tree. The nodes in the representation tree are the following polnomials: q (, ) =, q 2 (, ) = ( ) q (, ) =, q 3 (, ) = ( + ) q 2 (, ) = 2 2, q 4 (, ) = (2 ) q (, ) = 2. The polnomial of degree 3 is then a linear combination of nodes in the representation tree and coefficients f,..., f 4 which are polnomials of degree contained in the ellipses. This gives p(, ) = ( ) q (, ) + (2 + ) q 2 (, ) + ( + 3) q 3 (, ) + (2 ) q 4 (, ) = Similarl as in section 5, we can write the matrices b putting the linear coefficients in the first row and relations between the polnomials q (, ) to q 4 (, ) in the subsequent rows. For each edge of the form q k = (α + β) q j we put (α + β) in the position (k, j) in the matri M(, ) = A + B + C and in the position (k, k). In the first row we put a k + b k + c k in the position (, k) if f k (, ) = a k + b k + c k is the linear factor that multiplies the polnomial q k (, ) in the linearization. The matri M(, ) that corresponds to Figure 5, such that det(m(, )) = p(, ), is M(, ) = In Eample 6. we showed how to construct the bivariate pencil M(, ) = A + B + C from a representation tree and the corresponding linear coefficients. The outline of an algorithm that constructs a representation tree and the corresponding linear coefficients for a given polnomial p(, ) is presented in Algorithm. In the following discussion we give some missing details and show that the algorithm indeed gives a linearization. The nodes q 2,..., q n that we construct in step 2 are polnomials of the form q k (, ) = ( ζ ) ( ζ k ) for k = 2,..., n. All monomials in q k have degree k and the leading term is k.

14 A778 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH Algorithm. Given a bivariate polnomial p(, ) = α 00 +α 0 +α 0 + +α n0 n + α n, n + + α 0n n such that α n0 0, the algorithm returns a representation tree with a determinantal representation of the polnomial.. Compute n zeros ζ,..., ζ n of the polnomial h(t) = α n0 t n + α n, t n + + α 0n. 2. Form a branch of the tree with the root q (, ) and nodes q 2,..., q n, where q k+ is a successor of q k and the edge from q k to q k+ contains the factor ζ k for k =,..., n. 3. Compute linear coefficients f,..., f n for nodes q,..., q n in the following wa: (a) take f (, ) = α 00 + α 0 + α 0, (b) take f k (, ) = α k0 + (α k, α k0 β k ), where β k is a coefficient of q k (, ) at k, for k = 2,..., n, (c) take f n (, ) = α n0 ( ζ n ). 4. Compute the remainder r(, ) = p(, ) n i= f k(, ) q k (, ), which has the form r(, ) = 2 s(, ), where s(, ) is a polnomial of degree n If s(, ) 0, then stop and return the tree. 6. Add node q n+ and connect it to the root b an edge having the factor. 7. If s(, ) is a nonzero constant β 00, then use f n+ = β 00 as a coefficient for the node q n+, stop, and return the tree. 8. Recursivel call the same algorithm to obtain a representation tree with the root q for the polnomial s(, ). 9. Connect q n+ to q b an edge with a factor and return the tree with the root q. Each product q k (, )f k (, ) for k = 2,..., n is a polnomial with monomials of eact degree k, while q (, )f (, ) is a polnomial of degree. The linear factors f k (, ) in step 3 are constructed so that The leading two monomials ( k and k ) of f k (, ) q k (, ) agree with the part α k0 k + α k, k of the polnomial p(, ) for k = 2,..., n, the product f n (, ) q n (, ) = a n0 ( ζ ) ( ζ n ) agrees with the part of p(, ) composed of all monomials of degree eactl n, the product q (, ) f (, ) = a 00 + a 0 + a 0 agrees with the part of p(, ) composed of all monomials of degree up to. As a result, the remainder in step 4 has the form 2 s(, ), where s(, ) is a polnomial of degree n 3. The situation at the end of step 4 is presented in Figure 6. If the coefficient α n0 is zero, then we can appl a linear substitution of and of the form = and = ỹ + γ, where we pick γ such that α n, γ + α n 2,2 γ α 0n γ n 0. This ensures that the substituted polnomial in and ỹ will have a nonzero coefficient at n. After we complete the representation tree for the substituted polnomial in and ỹ, we perform the substitution back to and. If the polnomial s(, ) in step 4 is not a constant, then we obtain a representation subtree for s(, ) b calling recursivel the same algorithm. In order to obtain the final representation tree, we then join the eisting branch

15 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A779 f f n f 2 q 2 ζ 2 ζ f n q n q n ζ n ζ n 2 Fig. 6. The representation tree after step 4 of Algorithm. The remainder p(, ) n j= f j(, ) q j (, ) is a polnomial of the form 2 s(, ), where s(, ) is a polnomial of degree n 3. to the representation subtree for the polnomial s(, ). We do this b introducing a new node q n+ in step 6 that is linked to the root b the edge with the factor. To this new node we link the root q of the subtree for the polnomial s(, ) in step 9, again using the edge with the factor. As q is linked to the root b two edges, this multiplies all nodes in the subtree b 2 and, since the subtree is a representation for s(, ), this gives a representation for the remainder r(, ) from step 4. The situation after step 9 with the final representation tree for the polnomial p(, ) is presented in Figure 7. ζ ζ 2 q 2 q n+ ζ n 2 q ζ n q n q n subtree for s(, ) Fig. 7. The final representation tree. From the output of Algorithm, matrices A, B, C such that det(a+b+c) = p(, ) can be obtained in the same wa as in Eample 6.. We remark that the zeros ζ,..., ζ n in step can be comple, even if the polnomial p has real coefficients. Thus, in a general case a linearization produced b Algorithm has comple matrices A, B, and C.

16 A780 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH θ() = θ(2) = 2 θ(3) = 4 θ(4) = 6 θ(5) = 8 θ(6) = θ(7) = 4 θ(8) = 7 Fig. 8. Representation trees for polnomials of degrees to 8. Eample 6.2. We appl Algorithm on p(, ) = from Eample 5.3. First, we compute the roots (6.) ζ = i, ζ 2 = i, ζ 3 =.269 of the polnomial h(t) = 7t 3 + 8t 2 + 9t + 0. The zeros are ordered so that ζ ζ 2 ζ n. In eact computation the order is not important, but in numerical tests we eperience better results with this order. This gives the polnomials in the first branch of the representation tree: q (, ) =, q 2 (, ) = + ( i), q 3 (, ) = , and we can compute the corresponding coefficients f (, ) = +2+3, f 2 (, ) = 4+( i), f 3 (, ) = For the remainder r(, ) = p(, ) 3 j= f j(, ) q j (, ) = ( i) 2 we need just one additional node q 4 (, ) = with the coefficient f 4 (, ) = ( i). The determinantal representation with 4 4 matrices is p(, ) = det(a + B + C), where A = 0 0 0, B = 0 0 0, and i i C = i i Representation trees for polnomials of degree to 8 are presented in Figure 8. If we compare them to the determinantal representations from section 5 in Figure 4, then we see that representations obtained b Algorithm are much smaller. The following lemma shows that asmptoticall we use 3 fewer nodes than in section 5. Lemma 6.3. Algorithm returns representation tree G for the linearization of a polnomial p(, ) of degree n of size (6.2) θ(n) = G = { 6 n(n + 5), n = 3k or n = 3k +, 6 n(n + 5) 3, n = 3k + 2.

17 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A78 Proof. It follows from the recursion in the algorithm (see Figure 7) that the number of nodes satisfies the recurrence equation θ(n) = n + + θ(n 3). The solution of this equation with the initial values θ() =, θ(2) = 2, and θ(3) = 4 is (6.2). For generic polnomials of degrees n = 3 and n = 4 it turns out to be possible to modif the construction and save one node in the representation tree. The main idea is to appl a linear substitution of variables and in the preliminar phase to the polnomial p(, ) to eliminate some of the terms. This implies that the resulting matrices are of orders 3 (n = 3) and 5 (n = 4), instead of orders 4 and 6 as seen before, and it also reduces the size of the matrices for n = 3k and n = 3k + b. We give details in the following two subsections. 6.. The special case n = 3. We consider a cubic bivariate polnomial p(, ) = α 00 + α 0 + α α α 03 3, where we can assume that α We introduce a linear substitution of the form = + sỹ + t and = ỹ, where s is such that (6.3) h(s) := α 30 s 3 + α 2 s 2 + α 2 s + α 03 = 0 and t = α20s2 +α s+α 02 h (s). The substitution is well defined if s is a single root of (6.3) and the onl situation where this is not possible is when h has a triple root. This substitution transforms p(, ) into a polnomial p(, ỹ) such that its coefficients α 03 and α 02 are both zero. If we appl Algorithm to p(, ỹ) and choose ζ = 0 for the first zero, then the remainder in step 4 is zero and we get 3 3 matrices Ã, B, and C such that det(ã + B + ỹ C) = p(, ỹ). Now, it is eas to see that for A = Ã t B, B = B, and C = C s B, det(a + B + C) = p(, ). Eample 6.4. We take the recurrent eample p(, ) = (see Eamples 5.3 and 6.2). If we take s =.269 (see (6.3)) and t = , then substitution = + sỹ + t and = ỹ changes p(, ) into a polnomial p(, ỹ) = ỹ ỹ ỹ ỹ 2. Algorithm gives 3 3 matrices Ã, B, and C such that det(ã + B + ỹ C) = p(, ỹ), from which matrices [.0307 ] [ ] A = , B = 0 0, and [ ] i C = , i 0 such that det(a + B + C) = p(, ), are obtained and we have a 3 3 linearization.

18 A782 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH 6.2. The special case n = 4. Before we give a construction for a generic quartic bivariate polnomial, we consider a particular case when a polnomial p(, ) = 4 j=0 4 j k=0 α jk j k of degree 4 is such that α 30 = α 40 = α 03 = α 04 = 0. In this case five nodes are enough to represent the polnomial p(, ). The representation tree for the polnomial p(, ) is presented in Figure 9, where ζ and ζ 2 are the zeros of α 3 ζ 2 + α 22 ζ + α 3. α 00 +α 0 +α 0 α 20 + α α 02 α 2 + α 2 α 3 ( ζ 2 ) ζ ( ζ ) Fig. 9. A representation tree and a linearization for a polnomial p(, ) = 4 j=0 4 j k=0 α jk j k of degree 4 such that α 30 = α 40 = α 03 = α 04 = 0. For a generic quartic polnomial we first transform it into one with zero coefficients at 3, 4, 3, and 4. Ecept for ver special polnomials, we can do this with a combination of two linear substitutions. Similarl as in case n = 3, we first introduce a linear substitution of the form = + sỹ + t and = ỹ, where s is such that h(s) := α 40 s 4 + α 3 s 3 + α 22 s 2 + α 3 s + α 04 = 0 and t = α30s3 +α 2s 2 +α 2s+α 03 h (s). The substitution is well defined if s is a single root of h(s). After the substitution we have a polnomial p(, ỹ) such that its coefficients α 04 and α 03 are both zero. On this polnomial we appl a new substitution = and ỹ = u + ŷ + v, where g(u) := α 40 + α 3 u + α 22 u 2 + α 3 u 3 = 0 and v = α30+ α2u+ α2u2 g (u). This substitution is well defined if u is a single root of g(u); therefore, both substitutions eist for a generic polnomial of degree 4. After the second substitution we get a polnomial p(, ŷ) such that its coefficients α 30, α 40, α 03, and α 04 are all zero. For such a polnomial we can construct a representation with matrices 5 5 as presented in Figure 9. This gives 5 5 matrices B, Â, and Ĉ such that det(â + B + ŷĉ) = p(, ŷ). Finall, if we take A = Â t B (v tu) Ĉ, B = B u Ĉ, C = Ĉ s B, then det(a + B + C) = p(, ). If we add the constructions from subsections 6. and 6.2 as special cases to Algorithm, then we save one node for all generic polnomials of degree n = 3k or n = 3k +. Although this advantage seems to be modest, numerical results in the

19 ROOTS OF BIVARIATE POLYNOMIAL SYSTEMS A783 following section point out that for small n this does speed up the computation of the zeros considerabl (for instance, for n = 6 the corresponding -matrices are of order 0 2 = 00 instead of 2 = 2). 7. Numerical eamples. Determinantal representations from sections 5 and 6 can be used to numericall solve a sstem of two bivariate polnomials. We first linearize the problem as a two-parameter eigenvalue problem and then solve it with the method for singular two-parameter eigenvalue problems from [33], which is implemented in [39]. Subsequentl, we refine the solutions b two steps of Newton s method. We refer to the numerical methods that use linearizations from sections 5 and 6 as Lin and Lin2, respectivel. In the first eample we take polnomials with random coefficients, while the second eample considers some challenging benchmark polnomials. The following numerical eamples were obtained on a 64-bit Windows version of MATLAB R205a running on an Intel Core i GHz processor with 6 GB of RAM. Eample 7.. We compare Lin and Lin2 to the following methods: (a) NSolve in Mathematica 9 [5], (b) PHCLab.04 [2] running PHCpack , (c) BertiniLab.4 [37] running Bertini.5 [3], (d) NAClab 3.0, a MATLAB toolbo for numerical algebraic computation [52], where the last three methods use homotop. We compare methods on sstems of full bivariate polnomials of the same degree, whose coefficients are random real numbers uniforml distributed on [0, ] or random comple numbers, such that real and imaginar parts are both uniforml distributed on [0, ]. We also tested NSolve in Mathematica 0., but we do not report the results, as it is slower than Mathematica 9 for polnomials of small degree. The results are presented in Table. For each n we run all methods on the same set of 50 random polnomial sstems and measure the average time. Lin and Mathematica s NSolve work faster for polnomials with real coefficients, while this does not make a difference for Lin2, PHCLab, BertiniLab, and NAClab. Therefore, the results in the table for Lin2 and PHCLab are an average of 50 real and 50 comple eamples. Clearl, if Lin is applied to a polnomial with real coefficients, then matrices 0,, and 2 are real. If we appl Lin2, then the matrices are comple in general as roots of univariate polnomials are used in the construction. Although the comple arithmetic is more epensive than the real one, comple eigenproblems from Lin2 are so small that the are solved faster than the larger real problems from Lin. The sizes of -matrices obtained in Lin and Lin2 are presented in Table 2. Computational times for all methods ecept NSolve are ver similar for each of the 50 test problems of the same degree, with the small eception that Lin and Lin2 failed to compute solutions in 5 and 6 of the 800 sstems, respectivel, and were therefore successfull restarted with polnomials where variables and are interchanged. On the other hand, NSolve needs substantiall more time for certain sstems. For eample, for comple polnomials of degree n = 7, NSolve needed approimatel 0.65 s for 44 of the 50 eamples and 3.8 s for the additional 6 eamples. That eplains wh the average time for NSolve (C) in case n = 7 is just slightl smaller than in case n = 8. Lin is competitive in particular for real sstems of degree n 7, while Lin2 is the fastest method for real or comple sstems of degree n 9. For n 0 PHCLab becomes the fastest method.

20 A784 BOR PLESTENJAK AND MICHIEL E. HOCHSTENBACH Table Average computational times in milliseconds for Lin, Lin2, NSolve, and PHCLab for random full bivariate polnomial sstems of degree 3 to 0. For Lin and NSolve results are separated for real (R) and comple polnomials (C). Notice that these are the running times; the accurac of the methods varies, as we discuss in the tet. n Lin (R) Lin (C) Lin2 PHCLab NSolve(C) NSolve(R) BertiniLab NAClab Table 2 Size of -matrices for Lin and Lin2. Method n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 0 Lin Lin Beside the computational time, accurac and reliabilit are other important factors. The onl methods that return all solutions in all eamples are Lin, Lin2, and NSolve, but the results of NSolve are on average much less accurate than the results of all other methods. As a measure of accurac we use the maimum value of (7.) ma( p ( 0, 0 ), p 2 ( 0, 0 ) ) J ( 0, 0 ), where J( 0, 0 ) is the Jacobian matri of p and p 2 at ( 0, 0 ), over all computed zeros ( 0, 0 ). J ( 0, 0 ) is an absolute condition number of a zero ( 0, 0 ) and we assume that in random eamples all zeros are simple. For a good method (7.) should be as small as possible. For degree n 9, Lin2 is the fastest method and usuall also a ver accurate one. It is never significantl less accurate than the others, so it clearl wins in this case. The methods that are based on the homotop sometimes fail to compute all the solutions. This happens to BertiniLab, PHCLab, and NAClab in, 5, and 54 of 800 sstems. As the methods are using random initial sstems, a possible remed is to run them again. We remark that ever one fewer node in the representation tree reall makes a difference. For instance, if we do not appl the special case for n = 4 in subsection 6.2, then the -matrices for Lin2 for polnomial sstems of degree n = 0 are of size instead of and the average computational time rises from 0.63 s to 0.72 s. Eample 7.2. We test Lin and Lin2 on 25 eamples e00 to e025 from [6]. This set contains challenging benchmark problems with polnomials of small degree from (3, 2) to (, 0) that have man multiple zeros and usuall have fewer solutions than a generic pair of the same degrees. Lin and Lin2 performed satisfactoril on most eamples, but, using default parameters the also failed on some. Instead of giving the details for all 25 eamples, we give the ke observations. Multiple zeros can present a problem for the algorithm from [7] that is used to solve the projected regular problem w = 0 w, 2 w = 0 w that is obtained from (4.2) b the modified staircase algorithm from [33]. The