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Moment Matrices, Trace Matrices and the Radical of Ideals Moment Matrices, Trace Matrices and the Radical of Ideals y z Itnuit Janovitz-Freireich, Bernard Mourrain Lajos Rónyai ∗ Ágnes Szántó GALAAD MTA SZTAKI INRIA, 1111 Budapest, North Carolina State University Sophia Antipolis, L´agym´anyosi u. 11 Campus Box 8205 France Hungary Raleigh, NC, 27695 [email protected] [email protected] USA fijanovi2, [email protected] ABSTRACT 1. INTRODUCTION Let f1; : : : ; fs 2 K[x1; : : : ; xm] be a system of polynomials This paper is a continuation of our previous investigation generating a zero-dimensional ideal I, where K is an arbi- in [22, 23] to compute the approximate radical of a zero trary algebraically closed field. Assume that the factor alge- dimensional ideal which has zero clusters. The computation bra A = K[x1; : : : ; xm]=I is Gorenstein and that we have a of the radical of a zero dimensional ideal is a very important bound δ > 0 such that a basis for A can be computed from problem in computer algebra since a lot of the algorithms multiples of f1; : : : ; fs of degrees at most δ. We propose a for solving polynomial systems with finitely many solutions method using Sylvester or Macaulay type resultant matrices need to start with a radical ideal. This is also the case of f1; : : : ; fs and J, where J is a polynomial of degree δ gen- in many numerical approaches, where Newton-like methods eralizing the Jacobian, to compute moment matrices, and in are used. From a symbolic-numeric perspective, when we particular matrices of traces for A. These matrices of traces are dealing with approximate polynomials, the zero-clusters in turn allow us to compute a system of multiplicationp ma- create great numerical instability, which can be eliminated by computing the approximate radical. trices fMxi ji = 1; : : : ; mg of the radical I, following the approach in the previous work by Janovitz-Freireich, R´onyai The theoretical basis of the symbolic-numeric algorithm and Sz´ant´o. Additionally, we give bounds for δ for the case presented in [22, 23] was Dickson's lemma [14], which, in the m when I has finitely many projective roots in P . exact case, reduces the problem of computing the radical of K a zero dimensional ideal to the computation of the nullspace Categories and Subject Descriptors of the so called matrices of traces (see Definition 14): in [22, 23] we studied numerical properties of the matrix of I.1.2 [Computing Methodologies]: Algebraic Manipula- traces when the roots are not multiple roots, but form small tions|Algebraic Algorithms clusters. Among other things we showed that the direct computation of the matrix of traces (without the computa- General Terms tion of the multiplication matrices) is preferable since the Algorithms, Theory matrix of traces is continuous with respect to root pertur- bations around multiplicities while multiplication matrices Keywords are generally not. It turns out that the computationally most expensive part Moment Matrices, Matrices of Traces, Radical Ideal, Solving of the method in [22, 23] is the computation of the matrix of polynomial systems traces. We address this problem in the present paper, and ∗Affiliation: North Carolina State University, Department give a simple algorithm using only Sylvester or Macaulay of Mathematics. Research supported by NSF grant CCR- type resultant matrices and elementary linear algebra to 0347506. compute matrices of traces of zero dimensional ideals sat- yAffiliation: GALAAD, INRIA, Sophia Antipolis, France. isfying certain conditions. Research partially supported by the ANR GECKO More precisely, we need the following assumptions: let f = zAffiliation: Computer and Automation Institute of the [f1; : : : ; fs] be a system of polynomials of degrees d1 ≥ · · · ≥ Hungarian Academy of Sciences, and Budapest University ds in K[x], with x = [x1; : : : ; xm], generating an ideal I in of Technology and Economics. Research supported in part K[x], where K is an arbitrary algebraically closed field. We by OTKA grant NK63066. assume that the algebra A := K[x]=I is finite dimensional over K and that we have a bound δ > 0 such that a basis S = [b1; : : : ; bN ] of A can be obtained by taking a linear basis of the space of polynomials of degree at most δ factored by the subspace generated by the multiples of f ; : : : ; f of Permission to make digital or hard copies of all or part of this work for 1 s personal or classroom use is granted without fee provided that copies are degrees at most δ. By slight abuse of notation we denote not made or distributed for profit or commercial advantage and that copies the elements of the basis S which are in A and some fixed bear this notice and the full citation on the first page. To copy otherwise, to preimages of them in K[x] both by b1; : : : ; bN . Thus we can republish, to post on servers or to redistribute to lists, requires prior specific assume that the basis S consists of monomials of degrees at permission and/or a fee. Pm+1 most δ. Note that we can prove bounds δ = di − m Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. i=1 Pm (or δ = i=1 di − m if s = m) if I has only finitely many The motivation for this work was the papers [31, 32] where projective common roots in m and have no common roots they use moment matrices to compute the radical of real and PK at infinity, using a result of Lazard [33] (see Theorem 3). complex ideals. They present two versions of the method for Furthermore, we also assume that A is Gorenstein over K the complex case: first, in [32] they double up the machinery (see Definition 1). Note that in practice we can easily detect for the real case to obtain the radical of the complex ideal. if A is not Gorenstein (see Remark 10). Also, a random However, in [31] they significantly simplify their method and change of projective variables can eliminate roots at infinity show how to use moment matrices of maximal rank to com- with high probability when they are in finite number, but we pute the multiplicationp matrices of an ideal between I and will address the necessity of this assumption in an upcoming its radical I. In particular, in the Gorenstein case they paper. can compute the multiplication matrices of I. In fact, in The main ingredient of our method is a Macaulay type [31] they cite our previousp work [22] to compute the multi- resultant matrix Mac∆(f), which is defined to be a maximal plication matrices of I from the multiplication matrices of row-independent submatrix of the transpose matrix of the I, but the method proposed in the present paper is much Ps degree ∆ Sylvester map (g1; : : : ; gs) 7! i=1 figi 2 K[x]∆ simpler and more direct. for ∆ ≤ 2δ + 1 (see Definition 5). Using our assumptions Note that one can also obtain the multiplication matrices on A, we can compute a basis S of A using Mac∆(f), and of I with respect to the basis S = [b1; : : : ; bN ] by simply we also prove that a random element y of the nullspace eliminating the terms not in S from xkbi using Macδ+1(f). of Mac∆(f) provides a non-singular N × N moment matrix The advantagep of computing multiplication matrices of the MS (y) with high probability (similarly as in [31]). This mo- radical I is that it returns matrices which are always si- ment matrix allows us to compute the other main ingredient multaneously diagonalizable, and possibly smaller than the of our algorithm, a polynomial J of degree at most δ, such multiplication matrices of I, hence easier to work with. that J is the generalization of the Jacobian of f1; : : : fs in Moreover, if S contains the monomials 1; x1; : : : ; xm, one the case when s = m. The main result of the paper now can eigenvector computation yield directly the coordinates of the be formulated as follows: roots. Computation of the radical of zero dimensional complex Theorem Let S = [b1; : : : ; bN ] be a basis of A with deg(bi) ≤ ideals is very well studied in the literature: methods most δ. With J as above, let SylS (J) be the transpose matrix of related to ours include [18, 5] where matrices of traces are PN PN the map i=1 cibi 7! J · i=1 cibi 2 K[x]∆ for ci 2 K. used in order to find generators of the radical, and the ma- Then trices of traces are computed using Gr¨obner Bases; also, in [1] they use the traces to give a bound for the degree of the [T r(b b )]N = Syl (J) · X; i j i;j=1 S generators of the radical and use linear solving methods from there; in [19] they describe the computation of the radical where X is the unique extension of the matrix MS (y) such using symmetric functions which are related to traces. One that Mac∆(f) · X = 0: of the most commonly quoted method to compute radicals N is to compute the projections I\ [x ] for each i = 1; : : : ; m Once we compute the matrix of traces R := [T r(bibj )] K i i;j=1 and then use univariate squarefree factorization (see for ex- and the matrices R := [T r(x b b )]N = Syl (x J) · X xk k i j i;j=1 S k ample [17, 26, 10, 20] ). The advantage of the latter is that it for k = 1; : : : ; m, we can use the results of [22, 23] to compute can be generalized for higher dimensional ideals (see for ex- a system of multiplication matrices for the (approximate) ample [25]).
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