The subspace arrangement is = w here are linear subspaces of .

1. Relation between solvability of some multivariate interpolation problems and the variety of subspace arrangement. The subspace arrangement is = where are linear subspaces of . = }=d-s Algebraic geometry studies the property

2. George David Birkhoff Birkhoff Interpolation (in one variable) Birkhoff interpolation is an extension of Hermiteinterpolation. It involves matching of values and derivatives of a function at certain points without the requirement that the derivatives are consecutive. Example: find a polynomial If this equations have unique solution for all distinct and all the problem ( is regular. det =𝞅(,) ? ) :𝞅(, )}

3. Example: find a polynomial , det the problem ( is regular, completely regular. ) :𝞅(, )=0}= =

4. Birkhoff Interpolation problem: , where ; is regular if for any set of k district points and any f there exists unique p such that )=) for all k and all j=1,…,k. Isaac Schoenberg George Pólya The problem , is regular if k=n+1, ={0}, Lagrange interpolation For k>0, k, Hermite interpolation , ( =).

5. =(,=(, 𝞅(, )==𝞅(, , ) ; dim =3 if 𝞅 : } dim =2 If the scheme is regular then George G. Lorentz If 𝞅=3 If 𝞅 and then the determinant contains two identical rows (columns), hence The same is true in real case

6. Birkhoff Interpolation (in several variables) Carl de Boor Given a subspace a collection of subspaces , a set of points and a function f] we want to find a polynomial p such that ()f()=()f() for all q The scheme (, is regular if the problem has a unique solutions for all f] and all distinct , it is completely regular if it has unique solution for all . Amos Ron

7. Birkhoff Interpolation (in several variables) 𝞅(,…,) is a polynomial in = dim =max{dim}=kd-d If 𝞅=dk-1= Theorem: If (, is regular then Conjecture: it is true in the real case False A. Sharma Rong-Qing Jia

8. Haar subspaces and Haar coverings AlfrédHaar Definition: H=span { the determinant (The Lagrange interpolation problem is well posed) [x]. For d>1, n>1 there are no n-dimensional Haar subspaces in or even in C() J. C. Mairhuber in real case and I. Schoenberg in the complex case;. or from the previous discussion: (,{0},{0},…,{0})

9. Definition: A family of n-dimensional subspaces { the Lagrange interpolation problem is well posed in one of these spaces. =span { { Question: What is the minimal number s:= of n-dimensional subspaces ? Kyungyong Lee Conjecture: :

10. Theorem (Stefan Tohaneanu& B.S.): The three spaces , It remains to show that no two subspaces can do the job. =span { I want show that there are three distinct points =0 ), ),

11. ={(): =0} (0,0,0,0) (0,0,1,1) (-1,-1,0,0) dim =2=dim Not every two-dimensional variety in can be formed as a set of common zeroes of two polynomials. The ones that can are called (set theoretic) complete intersections. As luck would have it, is not a complete intersection: is not connected… Alexander Grothendieck Robin Hartshorne A very deep theorem states that a two dimensional complete intersection in can not be disconnected by removing just one point

12. =3 , The family of D-invariant subspaces spanned by monomials form a finite Haar covering. There are too many of them: for n=4 in 2 dimensions there are five: Yang tableaux =

13. köszönöm Thank You Thank You köszönöm