Carl de Boor. Introduction

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1 Multivariate polynomial interpolation: A GC 2 -set in R 4 without a maximal hyperplane Carl de Boor Abstract. A set T R d at which interpolation from Π n (R d ) (polynomials of degree n) is uniquely possible is a GC n-set if the associated Lagrange fundamental polynomials have only linear factors. For such GC n-sets T in the plane, Gasca and Maeztu conjectured the existence of a line containing n + 1 points from T. It is shown here that, already in R 4, there exist GC 2 -sets T without any hyperplanes containing dimπ 2 (R 3 ) points from T. Introduction This note is a follow-up on [B] inspired by Apozyan s work [A1], for which a convenient reference is [A2]. It concerns interpolation to data at some set T in R d from the set Π n = Π n (R d ) of polynomials of degree n in d variables. Call such a set T n-correct if the map δ T : p p T := (p(τ) : τ T), when restricted to Π n, is 1-1 and onto R T, i.e., if there exists, for every a R T, exactly one p Π n that agrees with a on T, i.e., satisfies p(τ) = a(τ), τ T. If #T = dimπ n (R d ), then the n-correctness of T follows already from δ T Π n (R d ) being 1-1 or onto. Among the n-correct sets, Chung and Yao [CY] have singled out what we now call GC n -sets. These are subsets T of R d (or C d ) with the property that, for every τ T, there exist n hyperplanes whose union contains all the points of T except τ. It is easily seen that a GC n -set is n-correct. In [GM], Gasca and Maetzu conjectured that every planar GC n -set has a maximal line, i.e., a straight line containing n+1 of the points of T. Such a line was called maximal in [B] since no line could contain more than n + 1 points from an n-correct planar set. I will refer to this as the GM n -conjecture. The conjecture has been verified only for n 5. In [CG], Carnicer and Gasca showed that if the GM n conjecture is true for all n n 0, then, for every n n 0, every planar GC n -set has three maximal lines. In an effort to understand better the GM n -conjecture, I proposed in [B] the following CG n -conjecture. Every GC n -set in R d has at least d + 1 maximal hyperplanes. Here, a hyperplane H in R d is called maximal for T if it contains the maximal number of points from T possible when T is n-correct. This number equals dimπ n (R d 1 ) = ( ) d 1+n n. Further, I disproved this conjecture by exhibiting in [B] a GC 2 -set in R 3 with only three maximal hyperplanes. The idea behind the construction of that counterexample has been used very cleverly by Apozyan in [A1] to construct a GC 2 -set in R 6 without any maximal hyperplanes, showing that the very GM-conjecture fails to hold for d 6. It is the purpose of this note to exhibit such a set in R 4 and, incidentally, show that the trivariate counterexample in [B] can be further refined into a GC 2 -set with just two maximal hyperplanes. 1

2 Construction of a GC 2 -set in R d Here is a well known recipe for obtaining a GC 2 -set in R d. Start with hyperplanes H 0,...,H d in R d in general position. This means that any d of them have exactly one point in common and that point is not on the remaining hyperplane. In symbols, H i {a i } := j i H j, i = 0,...,d. Now choose b ij on the line through a i and a j and different from either point. In symbols, b ij {a i, a j }\{a i, a j }, 0 i < j d. Note that b ij fails to lie in H i H j since, otherwise a i H i or a j H j, a contradiction. Let I claim that A := {a i : 0 i d}, B := {b ij =: b ji : 0 i < j d}. T := A B is a GC 2 -set. Indeed, for each 0 i d, H i contains all a j for j i, hence all b jk with j i k, making H i 2-maximal in the sense that #(T H i ) = dimπ 2 (R d 1 ). Further, (1) T\(H i {a i }) = {b ij : j i}, a set of d elements, hence contained in some hyperplane K i. However, this hyperplane cannot contain a i since, with each b ij it then would also have to contain a j, hence would have to contain A, therefore K i = H j, all j, contradicting the assumption that the hyperplanes H 0,...,H d are in general position. This shows that while (1) implies that T\(H i K i ) = {a i }, 0 i d, T\(H i H j ) = {b ij }, 0 i < j d. Therefore, T is a GC 2 -set. In particular, T is 2-correct. The move a i b ij b jk b ik b jk a j a k (2) Figure. The placement of the b jk and the move ijk. 2

3 Now consider the point set T obtained from T by one or more of the following kind of move : for a triple ijk with i j k i, we replace b jk by a point b jk in the punctuated flat {b ij, b ik }\{b ij, b ik }, making certain that no other move changes the points b ij, b ik. This ijk move takes place entirely within the plane P ijk := {a i, a j, a k }, hence entirely within the hyperplanes H h, h {i, j, k}, hence does not increase the number of points in any one hyperplane but does move one of the points from H i into K i, thus destroying the maximality of H i. We also require that at most one point is removed in this fashion from any one H i. Finally, note that neither b jk nor b jk are in H j H k, i.e., (3) {b jk, b jk } (H j H k ) =, as we observed earlier for b jk, to which we now add that if, e.g., b jk H k, then, as b ij H k, also b ik H k, therefore, as a i H k, also a k H k, a contradiction. Proposition. The set T resulting from such moves is again a GC 2 -set. Proof: We must, for each τ T, show T \{τ} to be contained in the union of two hyperplanes which does not contain τ. We follow [A1] by first proving that T is 2-correct and then proving that (4) τ T hyperplanes H τ, K τ, (T \{τ}) (H τ K τ ). The 2-correctness of T then implies that τ (H τ K τ ). τ T. T is 2-correct: Since #T = #T = dim Π 2 (R d ), it is sufficient to prove that δ T is 1-1 on Π 2 (R d ). For this, recall the observation that the move ijk takes place entirely within the plane P ijk = {a i, a j, a k }, and replaces one of the six points of T in P ijk by another point on that plane in such a way that the six points in P ijk of the resulting T form again a 2-correct set in that plane (the effect of removing b ij from {a j, a k } and placing its replacement b jk in {b ij, b ik }\{b ik, b ik }). Thus, if δ T p = 0 for some p Π 2 (R d ), then p must vanish on all of P ijk, and this holds for every P ijk with {i, j, k} a 3-subset of {0,..., d} whether or not a move took place in it, hence p must vanish on T, hence p = 0, by the 2-correctness of T. proof of (4) is by cases, and uses b jk to denote b jk or b jk as the case may be. Case τ = a i : If no point was removed from H i, then (5) T\(H i {a i }) = {b ij : j i}, a set of d elements, hence contained in some hyperplane, K i, proving (4) for this case. If, on the other hand, some point was removed from H i, then it was moved into the afore-mentioned K i, hence, either way, (T \{a i }) (H i K i ). Case τ = b jk : If all moves took place in H j or H k, then, by (3), T \(H j H k ) = {b jk }. Otherwise, any move not entirely within H j H k must involve both a j and a k, and, as b jk was not moved, must be a move jik or kij. Assume without loss of generality that it was the move jik. Then T \(H k {b jk }) = {b kr : r j, k} {a k }, a set of d points, hence contained in some hyperplane K k, proving (4) for this case. Case τ = b jk : Then, no other move could have involved j and k, hence, all other moves took place either entirely in H j or in H k. Therefore, with (3), T \(H j H k ) = {b jk }. 3

4 We now know that T is, indeed, a GC 2 -set, with certain H i now not maximal for T. Could our moves have generated a new maximal hyperplane for T different from the H i? To answer this question, we give an argument inspired by [A1]. Assume that H is such a hyperplane, and let µ := #(A H). Then µ < d since, otherwise, H = H i for some i (if µ = d), or H = R d (if µ = d + 1), and both of these possibilities are ruled out by our assumptions. Suppose that a j H while a k H. Then b jk H and if T contains instead b kj obtained through a move ijk then, as a k H, the line {a i, a k } has at most one point in common with H, hence the point b ik on that flat could be chosen so as not to lie in H, hence the point b jk can be chosen on the line {b ij, b ik } so as not to lie in H. Hence, either way, for every j, k with a j H a k, there is a point in T \H, with different pairs (j, k) giving rise to different points. Since there are µ(d+1 µ) such pairs, and in addition, #(A\H) = d + 1 µ, we get altogether at least (1 + µ)(d + 1 µ) points from T not in H while maximality of H requires #(T \H) to equal dimπ 2 (R d ) dimπ 2 (R d 1 ) = d + 1 which only leaves the conclusion that µ = 0. However, even if, at this point, H contained B = T \A, we could make that impossible in case d > 2 by changing one of the unmoved b jk from the unique point of intersection of {a j, a k } with H to any other point on that line (other than a j or a k ), thus making B = T \A not containable in a hyperplane. We haved proved the following Corollary. For d > 2, the (possibly slightly modified) GC 2 -set T has as maximal hyperplanes only those H i for which there was no move ijk. What is the maximal number of H i that could have lost their maximality? This will depend on d since, e.g., for d = 2, just one hyperplane H i can be deprived of a point by such a move but this happens only at the cost of generating a new maximal hyperplane, namely K i. a 2 a 3 a 0 (a) (b) (6) Figure. A 2-point variation (b) of a quadratic principal lattice (a) in R 3 has only 2 maximals. a 1 For d = 3, after the move 012, with b 12 now not in H 0, we cannot move the other two b-points within H 0 nor do we want to move them out of H 0. We are also obliged not to move the points b 01 and b 02, hence are left with just one other still movable point in B, namely b 03, which we could move within H 1 or H 2, thus moving it out of H 2 or H 1. Let s make the move 103, which requires us not to have moved b 01 nor b 13. The resulting T has only two of the H i still maximal. This is an improvement over the result in [B] where I made only the first move, hence retained 3 maximal hyperplanes. For d = 4, the move 012 will, as we saw in the case d = 3, permit, e.g., the move 103 within H 4, and these two moves remove b 12 from H 0 and b 03 from H 1. This ties down b 01, b 02 and b 13, leaving just b 23 in H 4 free to move, in the move 423 which removes b 23 from H 4 and ties down b 24 and b 34 but leaves b 04, b 14 4

5 free to move; both lie in H 2 H 3, suggesting the move 204 which removes b 04 from H 2 and/or the move 314 which removes b 14 from H 3. Could we do both moves? The move 204 requires us to keep b 02 and b 24, while the move 314 requires us to keep b 13 and b 34. Hence we can do both moves and, after those two moves, have altogether moved b 12 from H 0, b 03 from H 1, b 04 from H 2, b 14 from H 3, and b 23 from H 4 while keeping fixed b 01, b 02, b 13, b 24, b 34. Theorem. Already in R 4, there are GC 2 -sets without any maximal hyperplanes. Such a result was achieved by Apozyan in [A1] only in R 6 by a more cautious set of d + 1 moves any two of which had only one index in common, thereby ensuring that the b ij, b ik used in the move ijk will not be involved in any other move while it is only necessary to ensure that neither will be moved by another move; however, such a set of d + 1 cautious moves is only realizable for d > 5. o x x x o x x o x x o x x x o o x x o x x o x x x o x x o x x o x x x o (7) Figure. My moves vs Apozyan s moves. Acknowledgements. I acknowledge with pleasure the hours spent with Tomas Sauer in September 2011 working our way through the thesis [A1] and thank Hakop Hakopian for the kindness of providing me with a copy of it. My debts to Armen Apozyan are acknowledged throughout this note. References [A1] Armen Apozyan (2011), On multivariate polynomial interpolation, GC n -sets and Gasca-Maetzu conjecture, dissertation, Institute of Mathematics of the NAS RA, Yerevan (Armenia). [A2] Armen Apozyan (2011), A six-dimensional counterexample for the GM d conjecture, Jaen J. Approx. 3(2), [B] Carl de Boor (2008), Multivariate polynomial interpolation: conjectures concerning GC-sets, Numer. Algorithms 150(1), [CG] J. M. Carnicer and M. Gasca (2003), On Chung and Yao s geometric characterization for bivariate polynomial interpolation, in Curve and Surface Design: Saint-Malo 2002 (Tom Lyche, Marie-Laurence Mazure, and Larry L. Schumaker Eds.), Nashboro Press (Brentwood TN), [CY] K. C. Chung and T. H. Yao (1977), On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal. 14, [GM] M. Gasca and J. I. Maeztu (1982), On Lagrange and Hermite interpolation in IR k, Numer. Math. 39, sep12 5