On Primary Decomposition of Hermite Projectors

: An ideal projector on the space of polynomials C [ x ] = C [ x 1 , . . . , x d ] is a projector whose kernel is an ideal in C [ x ] . Every ideal projector P can be written as a sum of ideal projectors P ( k ) such that the intersection of their kernels ker P ( k ) is a primary decomposition of the ideal ker P . In this paper, we show that P is a limit of Lagrange projectors if and only if each P ( k ) is. As an application, we construct an ideal projector P whose kernel is a symmetric ideal, yet P is not a limit of Lagrange projectors.


Introduction
Let C[x] = C[x 1 , . . ., x d ] denote the algebra of polynomials in d variables with complex coefficients.A projector P on C[x] is a linear idempotent operator on C[x].Such a projector is called an ideal projector if ker P is an ideal in C[x].An ideal projector is called a Lagrange projector if ker P is a radical ideal in C[x].If the range of P is N-dimensional, then P is a Lagrange projector if and only if there exist N distinct points x 1 , . . ., x N ∈ C d such that ker P = { f ∈ C[x] : f (x 1 ) = . . .= f (x N ) = 0} or equivalently (P f )(x j ) = f (x j ) for all j = 1, . . ., N and all f ∈ C[x].The last equivalence shows that Lagrange projectors interpolate at nodes x 1 , ..., x N and therefore present a natural extension of the classical Lagrange interpolation theory to the multivariate setting.The notion of an ideal projector was first introduced by Birkhoff in [1].Since then, it was further studied, and connections to different branches of mathematics were explored (see [2][3][4][5][6][7][8][9][10][11]).In this paper, we consider exclusively finite dimensional ideal projectors.
In one variable, every Hermite interpolation projector is the limit of a sequence of classical Lagrange interpolation projectors.That allows us to extend the definition of the Hermite interpolation projectors to the multivariate setting as follows.
Definition 1.An ideal projector P is called a Hermite projector if there exist a sequence of Lagrange projectors P n on the range of P such that P n f → P f for every f ∈ C[x].We do not specify type of convergence because P n f and P f belong to the same finite-dimensional space; hence, all forms of convergence are equivalent.
In one variable setting, the ideal projectors are the same as classical Hermite projectors (see for example [10]).The natural question arises as to whether, in the multivariate setting, the same is true, i.e., is any ideal projector necessarily a limit of Lagrange projectors?Rather surprisingly, the resulting answer is positive in two variables (cf.[4]) but negative in three or more variables (cf.[12]).The question of a description of those ideal projectors that are Hermite was raised by Carl de Boor in [3].Some partial results regarding this question were obtained in [8,9] and, in the very different language of algebraic geometry, in [13,14].In this paper, we make a contribution to this problem by examining the primary decomposition of Hermite projectors.
Every finite-dimensional ideal projector P can be written as a finite (direct) sum of ideal projectors P (k) P = P (1) where P (k) are ideal projectors such that the ideals ker P (k) form the primary decomposition of the ideal ker P. That is and, for each P (k) , the variety consists of exactly one point.
The main result of this paper is Theorem 1. P is Hermite if and only if each P (k) is Hermite.
Based on the above theorem, as an application, we will show the existence of a symmetric ideal projector (in three or more variables) that is not Hermite (see Theorem 7).Finally, we will showcase some problems in matrix theory (see Problem 2) that are related to the main result.

Preliminaries
Let C [x] denote the algebraic dual of C[x], i.e., the space of all linear functionals on C[x].For an ideal J ⊂ C[x], let V (J) denote the affine variety associated with J: The ideal J has a finite codimension (0-dimensional) if and only if the set V (J) is finite (cf.[15]).Moreover, |V (J)| ≤ dim(C[x]/J) and |V (J)| = dim(C[x]/J) if and only if the ideal J is radical i.e., J = { f : f (x) = 0, for all V (J)}.
Let J ⊥ denote a subspace of C [x] of all functionals that vanish on J. Hence to denote the point evaluation functional: It is easy to see that for any N-dimensional Lagrange interpolation projector P, the variety V (ker P) is consisting of exactly N distinct points.Assuming V (ker P) = {x 1 , . . ., x N }, we have ker ⊥ P = span{δ x 1 , ..., δ x N }.
Below, we will review relations between ideal projectors and the sequence of commuting matrices.
Let P be an N-dimensional ideal projector and let G be its range.Hence, C[x] = G ⊕ ker P and ker P is an ideal of codimension N in C[x].Thus, C[x]/ ker P is an N-dimensional algebra.For each coordinate x j of x, we define a multiplication operator (N × N matrix) M j : C[x]/ ker P → C[x]/ ker P associated with P by where [•] represents a class equivalence in C[x]/ ker P. The set (M 1 , . . ., M d ) forms a sequence of commuting matrices that are associated with the projector P. In fact, such a sequence uniquely defines P (see [4] for details).The matrices M j represent the operators defined on G by M j ( f ) = P(x j f ).
Theorem 3 (cf.[4]).Suppose that we have a sequence of ideal projectors P n onto the same space G and let (M d ) be multiplication operators associated with P n while (M 1 , . . ., M d ) is the multiplication operators on G associated with P.Then, P n → P if and only if (M Next, we prove an extension of Theorem 5.2.1 in Artin [18] to the set of commuting matrices.Our proof is substantially different than the one presented there.Proof.Let (λ Assume without loss of generality that u n = 1.Then are uniformly bounded, which proves the first part of the theorem.Now, passing to a subsequence if necessary, we may assume that λ The sequence (u n ) is uniformly bounded in a finite-dimensional space G; hence, it is compact.Passing to a subsequence if necessary, we may assume that u n → u ∈ G and u = 1.Finally, since (M In addition, λ Combining the above with the Theorem 2, we obtain: Corollary 1.Let P be an N-dimensional Hermite projector and P n be a sequence of Lagrange projectors such that P n → P. If V (ker P) = {x 1 , x 2 , . . ., x N } and V (ker Then, there exists a constant C such that |x (n) k | ≤ C for all k and n.Additionally, x 1 , x 2 , . . ., x K are the only possible limit points of the set n V (ker P n ).Now, we will recollect a few facts regarding the convergence of ideal projectors.The main idea is that such a convergence depends only on their respective kernels.For more details and proofs, see [12].
Theorem 5 (cf. [12]).Let P n and P be ideal projectors onto a finite-dimensional space G ⊂ C[x].Then, P n → P if and only if for every functional F ∈ ker ⊥ P, there exists a sequence of functionals F n ∈ ker ⊥ P n such that F n → F in the weak-topology.i.e., (5) If each P n is a Lagrange projector, then ker ⊥ P n is spanned by N point evaluation functionals δ , and each F n can be written as their linear combination.Using the above theorem, we obtain the following.
Corollary 2. An N-dimensional ideal projector P is Hermite if and only if every F ∈ ker ⊥ P is the weak-limit of linear combinations of N point evaluation functionals.That is, there exists sets X n ⊂ C d , each consisting of N distinct points such that for every F ∈ ker ⊥ P for some coefficients a (n) x and for all f ∈ C[x].

The Main Result
The main goal is to prove Theorem 1.One side is easy to establish and can be shown as follows.Let P be an N-dimensional ideal projector and assume that ker P has a primary decomposition ker P = m j=1 J j where the ideals J j have codimensions equal to N j , respectively.Since ker ⊥ P = ⊕J ⊥ j we have ∑ M j=1 N j = N.Take any F ∈ ker ⊥ P.Then, F = ∑ m j=1 F j for some F j ∈ J ⊥ j .If each J j is the kernel of a Hermite projector then, by Corollary 2, there exists sets X j n ⊂ C d consisting of N j distinct points such and therefore, F is a weak-limit of a linear combination of N point evaluations.By Corollary 2, P is Hermite.
The main result of this section is a proof of the converse statement.The main idea is as follows.Assume P is Hermite, P = P (1) ⊕ P (2) ⊕ . . .⊕ P (m) and ker P has a primary decomposition ker P = ∩ m j=1 ker P (j) .By Corollary 2, there exists sets X n ⊂ C d such that (6) holds.Let V (ker P (1) ) = {y}.We will decompose X n = Y n ∪ Z n so that all accumulation points of n∈N Z n are away from y.For every functional F ∈ ker ⊥ P (in particular, every functional F ∈ ker ⊥ P 1 ), we have for some coefficients a x and for all f ∈ C[x].The main part of the proof is to show that the above implies that Thus, in (7), we can eliminate all point evaluations that do not accumulate at y, and the number of points remaining is equal to N 1 = dim ker ⊥ P 1 .Hence, by Corollary 2, the projector P 1 is Hermite.
To carry the proof in detail, we need a few preliminary results.First, we will produce a multivariate analog of Lagrange fundamental polynomials that seems to be new.Proof.Let < u, v > denote the Hermitian inner product in the space C d .Consider the following polynomial , ω(x) is a polynomial of degree at most m.Clearly, ω(y) = 0 for all y ∈Y and ω(z) = 1.
Since x, y, z lie in a ball of radius R m+1 → 1, we conclude that ∑ m+1 j=1 u (n) j → 0 as required.
We are now ready for the proof of the main theorem.
Theorem 6.Let P be a Hermite projector onto an N-dimensional space G ⊂ C[x].Suppose that P = P (1) ⊕ P (2) ⊕ . . .P (m) (11) where P (k) are ideal projectors such that the ideals ker P (k) form the primary decomposition of the ideal ker P Then, each P (k) is Hermite.
Since P is Hermite, by Corollary 2, for every functional for every f ∈ C[x].In particular, if F ∈ ∩ m−1 j=1 ker ⊥ P (j) , then due to (12), F ∈ ker ⊥ P and hence (13) holds.By Corollary 1, the sets X n lie in some ball in C d of radius R and {u 1 , . . .u m } are the only accumulation points of ∪ n X n .Partition the points X n = Y n ∪ Z n so that for every sequence x n ∈ Z n , we have and, for sufficiently large n, the points x n ∈ Y n are arbitrarily close to the set {u 1 , . . .u m−1 }.
In particular, x n − u m ≥ r > 0, for some r and for all x n ∈ Y n .
We rewrite (13) as where the points in Y n and Z n satisfy ( 15) and ( 14), respectively.Now, let p be a polynomial in ∩ m−1 j=1 ker P (j) such that p(u m ) = 1.Such a polynomial exists since, otherwise, every polynomial in ∩ m−1 j=1 ker P (j) would vanish at u m and hence u m ∈ V (∩ m−1 j=1 ker P for k = 1, . . ., m where ω Y n ,u m is defined as in Proposition 1, and f is arbitrary.Since p is in the ideal ∩ m−1 j=1 ker P (j) so are h k,n , hence F(h k,n ) = 0.By the same proposition and by ( 14), these polynomials are uniformly bounded and belong to a finite-dimensional space of polynomials of degree ≤ (mm + deg p)+ deg f .Thus, the convergence (16) on this space is uniform, and (16) gives x h k,n (x) = 0, (19) ing question about the extension of this result to an arbitrary commuting block-diagonal sequence of matrices.
Problem 2. Let M =(M 1 , . . ., M d ) be a sequence of commuting matrices such that each M j is block diagonal, i.e., of the form of (21).Let M (j) =(M (j) 1 , . . ., M d ) be of the same size and commute.Suppose that M is a limit of simultaneously diagonalizable matrices.Does it imply that each sequence M (j) is a limit of simultaneously diagonalizable matrices?Remark 1.In the language of algebraic geometry, the ideals that serve as kernels of Hermite projectors are called "smoothable" (cf.[14]).Hence, the main result of this paper formulated in this language says that an ideal J is smoothable if and only if every ideal in the primary decomposition of J is smoothable.

Proposition 1 .
Let Y be a finite set of m points in C d and take z ∈C d such that Y and z lie in the interior of a ball B ⊂ C d of radius R. Let r = min{ y − z : y ∈ Y } > 0.Then, there exists a constant C(R, r) and polynomial ω(x) = ω Y,z ∈ C[x] of degree at most m such that ω(z) = 1, ω(y) = 0, for all y ∈Y and ω B ≤ C(R, r) = 2R r 2m .(here, ω B denotes the supremum of the polynomial ω over the ball B ⊂ C d ).

Author Contributions:
Conceptualization, B.S., L.S. and B.T.; investigation, B.S., L.S. and B.T.; methodology, B.S., L.S. and B.T.; writing-original draft preparation, B.S., L.S. and B.T.; writing-review and editing, B.S., L.S. and B.T. All authors have read and agreed to the published version of the manuscript.Funding: This research received no external funding.