New Results Concerning the Gasca-Maeztu Conjecture Kurt Jetter Universit¨ at Hohenheim New Results Concerning the Gasca-Maeztu Conjecture – p. 1/36
Synopsis Introduction and Notation The Gasca-Maeztu Conjecture Maximal Lines and Maximal Curves n -dependent Sets Sets Satisfying the Chung-Yao Geometric Condition The m -distribution Sequence of a Node Lines Used by Several Nodes Some Ideas in the Proof for n = 5 Outlook Joint work with Hakop Hakopian and Georg Zimmermann New Results Concerning the Gasca-Maeztu Conjecture – p. 2/36
Notation – Π n the space of algebraic polynomials of (total) degree at most n ≥ 1 . ` n +2 ´ – X ⊂ R 2 with | X | = dim Π n = . 2 General Assumption: – X is n -poised ( n -regular, n -correct), i.e., and p ∈ Π n p ( x ) = 0 , x = ( x, y ) ∈ X = ⇒ p = 0 . – Equivalently, for any real (or complex) data { c x , x ∈ X } , there is a unique polynomial p ∈ Π n , interpolating these data: p ( x ) = c x , x ∈ X . – Equivalently, the Vandermonde with respect to X and any basis of Π n is regular. Here, in case we use the monomial basis, the Vandermonde V = V n ( X ) consists of rows [ · · · x α · · · ] α =( α 1 ,α 2 ) ∈ N 2 x α = x α 1 · x α 2 for 0 ,α 1 + α 2 ≤ n , x = ( x 1 , x 2 ) ∈ X , . 1 2 New Results Concerning the Gasca-Maeztu Conjecture – p. 3/36
Fundamental Polynomials n -poisedness is equivalent to the fact that for any x ∈ X , and the data e x = ( δ x , y ) y ∈ X , we have a unique (so-called Lagrange) fundamental polynomial p x ∈ Π n , determined by p x ( y ) = δ x , y , y ∈ X , (1) and the polynomial interpolating the data ( c x ) x ∈ X may be written (in its Lagrange form) as X p = c x p x . x ∈ X Remark. Fundamental polynomials satisfying (1) can be also considered for not necessarily n -poised sets X , in particular if | X | < dim Π n . In this case, for some point x ∈ X such a fundamental polynomial may not exist and/or may not be unique. New Results Concerning the Gasca-Maeztu Conjecture – p. 4/36
The Gasca-Maeztu Conjecture The Gasca-Maeztu conjecture (for short: GM-conjecture or GM n - conjecture ) is based on the assumption that each fundamental polynomial p x , x ∈ X , factors into linear polynomials. It states that under this assumption at least n + 1 nodes from the set X must be collinear . It was first stated in Gasca-Maeztu [2]; see also Carnicer-Gasca [3]. Surprisingly, up to now the conjecture is verified only for n ≤ 5 . n = 1 Here, | X | = dim Π 1 = 3 , and X is 1 -poised iff the three nodes are not collinear. For each node x ∈ X , the fundamental polynomial p x is (the linear polynomial whose zero set is) the line connecting the two other nodes from X . n = 2 Here, | X | = dim Π 2 = 6 . Since p x , for each x ∈ X , is the product of two lines containing five nodes from X , one of these lines must contain 3 nodes. The assumption of the GM-conjecture leads to sets X of the following type: The nodes lie on three lines, which are not concurrent, on each line three nodes, and the intersection points of any two lines belong to X . New Results Concerning the Gasca-Maeztu Conjecture – p. 5/36
The Gasca-Maeztu Conjecture, Proofs n = 3 Here, | X | = dim Π 3 = 10 . Assuming that GM 3 does not hold, we find that for each node x ∈ X , the fundamental polynomial p x factors into three lines each containing three nodes. Also the fundamental polynomials of two different nodes cannot share a common line. So, for the ten nodes in X , we need 30 different lines, each one containing three nodes of X . ` 10 ´ ` 3 ´ This is impossible, since there are at most = 15 such lines. / 2 2 n = 4 The first proof was given by Busch [4]. See also Carnicer and Gasca [3], or HJZ [5]. n = 5 This case is treated here; see HJZ [1]. New Results Concerning the Gasca-Maeztu Conjecture – p. 6/36
The Geometric Condition GC n Chung and Yao [6] have introduced this notion: An n -poised set X ⊂ R 2 satisfies the geometric condition GC n , if for each x ∈ X the set X \ { x } is contained in the union of n lines ℓ x n , say, 1 , . . . , ℓ x Γ x := { ℓ x 1 , . . . , ℓ x X \ { x } ⊂ n } . This is equivalent to the requirement that X is n -poised, and for each x ∈ X , the fundamental polynomial is the product of the n linear factors in Γ x : n Y ℓ x p x = γ x j . j =1 Here, γ x � = 0 is a normalization constant enforcing p x ( x ) = 1 . Since X is assumed to be poised, the set Γ x is uniquely determined by x , up to normalization of the lines equations. (Here, and in what follows, we identify a line ℓ with the linear (normalized) polynomial defining the line as its zero set.) New Results Concerning the Gasca-Maeztu Conjecture – p. 7/36
The Natural Lattices by Chung and Yao These so-called natural lattices are the most efficient point lattices satisfying GC n . Take n + 2 lines of R 2 in general position, i.e., – each pair of lines intersects at a single point, and – no triple of lines is concurrent. Then ´ intersection points of all pairs of lines is n -poised, and – the set X of the ` n +2 2 – for each x ∈ X , the set Γ x consists of those n lines not containing x . For this Chung-Yao natural lattice, each of the n + 2 lines contains exactly n + 1 nodes from X . New Results Concerning the Gasca-Maeztu Conjecture – p. 8/36
Example of a Chung and Yao Natural Lattice ( n = 3 ) New Results Concerning the Gasca-Maeztu Conjecture – p. 9/36
The Berzolari-Radon Type Sets X ` n +2 ´ Consider n + 1 different lines ℓ 1 , . . . , ℓ n +1 , and a set X of nodes with | X | = such 2 that, for each k = 1 , . . . , n + 1 , “ ” ∪ k − 1 n + 2 − k nodes of X are on ℓ k \ j =1 ℓ j . Then X is n -poised, cf. [9], [10]. Such a so-called Berzolari-Radon set X does not necessarily satisfy GC n . However, at least one point, namely the point x on ℓ n +1 not contained in the former n lines, has a fundamental polynomial which factors according to n Y p x = γ x ℓ j . j =1 New Results Concerning the Gasca-Maeztu Conjecture – p. 10/36
Example of a Berzolari-Radon Type Set X ( n = 3 ) ℓ 1 ℓ 2 ℓ 4 ℓ 3 New Results Concerning the Gasca-Maeztu Conjecture – p. 11/36
Point Distribution of a Berzolari-Radon Type Set In the former example for n = 3 , the fundamental polynomial of the last point x ∈ ℓ 4 is the product of the lines ℓ 1 , ℓ 2 , ℓ 3 which contain the remaining points X \ { x } according to the distribution sequence (4 , 3 , 2) . For general n , the last point x ∈ ℓ n +1 in the construction of a Berzolari-Radon set uses the line sequence ( ℓ 1 , ℓ 2 , ℓ 3 , . . . , ℓ n ) according to the point distribution sequence ( n + 1 , n, n − 1 , . . . , 2) . For the Chung-Yao natural lattices, for any x ∈ X , the used line sequence is the sequence of those n lines not containing x (in any order), and the corresponding point distribution sequence is again given by ( n + 1 , n, n − 1 , . . . , 2) . New Results Concerning the Gasca-Maeztu Conjecture – p. 12/36
Line Sequences and Distribution Sequences for x ∈ X Assumption. In what follows, we assume that X is an n -poised GC n -set. We use the following notations, for given x ∈ X : – Γ x = { ℓ x n } , the (unordered) set of lines (linear polynomials) used in the 1 , . . . , ℓ x factorization of the fundamental polynomial p x . This set is uniquely determined, up to normalization of the polynomials. Notation. We say that x uses the lines from Γ x . – ( ℓ 1 , . . . , ℓ n ) any ordered n -tuple of all lines ℓ x j ∈ Γ x , is called a line sequence for x . – ( k 1 , . . . , k n ) , the corresponding distribution sequence for x ∈ X , is determined by the following count of the nodes of X \ { x } : ˛ “ ”˛ ˛ ˛ k i = ˛ ℓ i ∩ ( X \ { x } ) \ ∪ j<i ℓ j ˛ , i = 1 , . . . , n . New Results Concerning the Gasca-Maeztu Conjecture – p. 13/36
Count of Intersection Points in X So, given a line sequence ( ℓ 1 , . . . , ℓ n ) for x ∈ X , k i is the number of nodes from X \ { x } lying on ℓ i , but not on the former lines ℓ 1 , . . . , ℓ i − 1 . Intersection points of two lines need special consideration: Notation. If { x } := ℓ i ∩ ℓ j ∈ X , for some i < j , and if x is counted by k i , then x is called a primary node for ℓ i and a secondary node for ℓ j . Fact. The distribution sequence of a node x ∈ X is uniquely determined when the line sequence is fixed, but not conversely, and we have ˛ ˛ ˛ . ˛ X \ { x } k 1 + · · · + k n = New Results Concerning the Gasca-Maeztu Conjecture – p. 14/36
A Point, and the Lines Used by It ( n = 5 ) ℓ 1 ℓ 2 A ℓ 3 ℓ 4 ℓ 5 New Results Concerning the Gasca-Maeztu Conjecture – p. 15/36
Maximal Distribution Sequence for x ∈ X For given x ∈ X , there are many line sequences and corresponding distribution sequences. Among the distribution sequences ( k 1 , . . . , k n ) for x , there is a unique maximal one (with respect to lexicographic ordering). We call this the maximal distribution sequence for x . It is ordered decreasingly according to n + 1 ≥ k 1 ≥ k 2 ≥ · · · ≥ k n ≥ 2 . There may be several line sequences leading to the same maximal distribution sequence of x . In the former example, the maximal distribution sequence is given by (5 , 5 , 4 , 3 , 3) , with several corresponding line sequences, e.g., or ( ℓ 3 , ℓ 1 , ℓ 2 , ℓ 5 , ℓ 4 ) , or · · · ( ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 , ℓ 5 ) , New Results Concerning the Gasca-Maeztu Conjecture – p. 16/36
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