A new study on the vanishing ideal of a set of points with - PowerPoint PPT Presentation

A new study on the vanishing ideal of a set of points with multiplicity structures Na Lei, Xiaopeng Zheng, Yuxue Ren School of Mathematics, Jilin University na.lei.cn@gmail.com 2012-10-27 A new study on the vanishing ideal of a set of points with multiplicity

Outline ♣ Problem description ♣ Ideas ♣ Examples ♣ Algorithms A new study on the vanishing ideal of a set of points with multiplicity

Problem description A new study on the vanishing ideal of a set of points with multiplicity

Lower set and limiting set D ⊆ N n 0 is called a lower set as long as whenever d = ( d 1 , . . . , d n ) lies in D and d i � = 0 , d − e i also lies in D where e i = ( 0 , . . . , 0 , 1 , 0 , . . . , 0 ) , and the 1 situated at the i -th position(1 ≤ i ≤ n ). For a lower set D , we define its limiting set E ( D ) to be the set of all β ∈ N n 0 − D such that whenever β i � = 0, then β − e i ∈ D . Fig.1: Illustration of three lower sets and their limiting sets. A new study on the vanishing ideal of a set of points with multiplicity

Points with multiplicity structures � p , D � represents a point p with multiplicity structure D , where p is a point in affine space A n and D is a lower set. ♯ D is called the multiplicity of point p . H = {� p 1 , D 1 � , . . . , � p t , D t �} is a set of points with multiplicity structures. A new study on the vanishing ideal of a set of points with multiplicity

The vanishing ideal of the set of points with multiplicity structures Given � p , D � , for each d = ( d 1 , . . . , d n ) ∈ D , we define a corresponding functional ∂ d 1 + ... + d n L ( f ) = f ( p ) . ∂ x d 1 1 . . . ∂ x d n n For H = {� p 1 , D 1 � , . . . , � p t , D t �} , we can define m functionals where m � ♯ D 1 + . . . + ♯ D t . Our aim is to find the reduced Gr ¨ o bner basis of the vanishing ideal I ( H ) = { f ∈ k [ X ]; L i ( f ) = 0 , i = 1 , . . . , m } under the lexicographic ordering with X 1 ≻ X 2 ≻ . . . ≻ X n . A new study on the vanishing ideal of a set of points with multiplicity

Related works o ller, T. Mora, Gr ¨ [1]M.G. Marinari, H.M. M - o bner bases of ideals defined by functionals with an application to ideals of projective points, J. AAECC 4 (2) (1993) 103-145. [2] L. Cerlinco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1995) 73-87. [3]Mathias Lederer, The vanishing ideal of a finite set of closed points in affine space, Journal of Pure and Applied Algebra 212 (2008) 1116-1133. A new study on the vanishing ideal of a set of points with multiplicity

What’s new? We solve the problem by induction over variables and introduce a new algorithm to compute the intersection of some special ideals. The algorithm has an explicit geometric interpretation which reveals the essential connection between the relative position of the points and the quotient basis of the vanishing ideal. The algorithm offers us a new perspective of view to look into the reduced Gr ¨ o bner basis. This new view can help us understand the problem better and obtain some new conclusions. A new study on the vanishing ideal of a set of points with multiplicity

Ideas A new study on the vanishing ideal of a set of points with multiplicity

Main ideas of our algorithm: Split the set of points into several subsets according to the coordinates of the smallest variable x n of the points. In each subset the points have the common value of x n . Compute the reduced Gr ¨ o bner basis and quotient basis of the vanishing ideal of each subset. Compute the intersection of the vanishing ideals of these subsets. A new study on the vanishing ideal of a set of points with multiplicity

Main ideas of our algorithm: Split the set of points into several subsets according to the coordinates of the smallest variable x n of the points. In each subset the points have the common value of x n . Compute the reduced Gr ¨ o bner basis and quotient basis of the vanishing ideal of each subset. Compute the intersection of the vanishing ideals of these subsets. A new study on the vanishing ideal of a set of points with multiplicity

Main ideas of our algorithm: Split the set of points into several subsets according to the coordinates of the smallest variable x n of the points. In each subset the points have the common value of x n . Compute the reduced Gr ¨ o bner basis and quotient basis of the vanishing ideal of each subset. Compute the intersection of the vanishing ideals of these subsets. A new study on the vanishing ideal of a set of points with multiplicity

Examples A new study on the vanishing ideal of a set of points with multiplicity

Two representing forms of points with multiplicity structures Given H = {� p 1 , D 1 � , � p 2 , D 2 � , � p 3 , D 3 �} , where p 1 = ( 1 , 1 ) , D 1 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) } p 2 = ( 2 , 1 ) , D 2 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) } p 3 = ( 0 , 2 ) , D 3 = { ( 0 , 0 ) , ( 1 , 0 ) } . A new study on the vanishing ideal of a set of points with multiplicity

    1 1 0 0     1 1 1 0         1 1 0 1         2 1 0 0         P = 2 1 , D = 1 0 .         2 1 0 1         2 1 1 1         0 2 0 0 0 2 1 0 A new study on the vanishing ideal of a set of points with multiplicity

Example 1: Vanishing ideal of the subset. (All the points share the same X n coordinates) Given H = {� p 1 , D 1 � , � p 2 , D 2 �} , where p 1 = ( 1 , 1 ) , D 1 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) } , p 2 = ( 2 , 1 ) , D 2 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) } . All the points share the same X 2 coordinates 1, denote the greatest value in the last column of matrix D as w . A new study on the vanishing ideal of a set of points with multiplicity

Group the row vectors of matrix P and D according to the values in the last column of matrix D . • case 1:     1 1 0 0     1 1 1 0     P 1 =  , D 1 =  .   2 1 0 0 2 1 1 0 • case 2:     1 1 0 1   , D 2 =   . P 2 = 2 1 0 1 2 1 1 1 A new study on the vanishing ideal of a set of points with multiplicity

• case 1: Denote by a the value in the last column of matrix P 1 , and b for that of D 1 , that is a := 1 , b := 0. And then eliminate the last columns of matrices P 1 and D 1 to get an univariate problem which can be solved according to the induction assumption. Quotient basis: ˆ D 1 := { 1 , X 1 , X 2 1 , X 3 1 } . o bner basis: ˆ G 1 := { ( X 1 − 1 ) 2 ( X 1 − 2 ) 2 } . Gr ¨ D 1 ) , ˜ G 1 := mapg a , b ( ˆ D 1 := embed b (ˆ ˜ G 1 ) . ′ ∈ N n − 1 ′′ ∈ N n embed c : D − → D 0 , 0 ( d 1 , . . . , d n − 1 ) − → ( d 1 , . . . , d n − 1 , c ) . ′ ( n − 1-variable polynomial set) − ′′ mapg a , b : G → G ( n -variable polynomial set) → g · ( X n − a ) b . g − A new study on the vanishing ideal of a set of points with multiplicity

• case 2: a := 1 , b := 1. In the same way, we have ˜ D 2 and ˜ G 2 . • Unit the results of case 1 and case 2: � ˜ D := ˜ D 1 D 2 is the quotient basis. � ˜ � G := ˜ { ( X 2 − a ) w + 1 } G 1 G 2 is the reduced Gr ¨ o bner basis. A new study on the vanishing ideal of a set of points with multiplicity

The geometric interpretation of the the algorithm. A new study on the vanishing ideal of a set of points with multiplicity

Example 2: Vanishing ideal of arbitrary set of points. (Intersection of two ideals) Given H = {� p 1 , D 1 � , � p 2 , D 2 � , � p 3 , D 3 �} , where p 1 = ( 1 , 1 ) , D 1 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) } , p 2 = ( 2 , 1 ) , D 2 = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) } , p 3 = ( 0 , 2 ) , D 3 = { ( 0 , 0 ) , ( 1 , 0 ) } . A new study on the vanishing ideal of a set of points with multiplicity

Group the points into two sets according to the X 2 coordinates (the values in the last column of matrix P ). Each can be solved as Example 1. A new study on the vanishing ideal of a set of points with multiplicity

Intersection of two ideals h 1 = ( X 2 − 1 ) 2 ; g 1 = X 2 − 2 ; h 2 = ( X 2 − 1 )( X 1 − 1 )( X 1 − 2 ) 2 ; g 2 = X 2 1 ; h 3 = ( X 1 − 1 ) 2 ( X 1 − 2 ) 2 . f 1 := g 1 · h 1 . A new study on the vanishing ideal of a set of points with multiplicity

• To compute f3 whose leading term should be X 3 1 X 2 : g 3 := X 1 · g 2 , the leading terms of g 3 and h 2 share the same degree of X 1 with X 3 1 X 2 as a factor. q 1 := h 2 · g 1 = ( X 2 − 1 )( X 1 − 1 )( X 1 − 2 ) 2 ( X 2 − 2 ) , q 2 := g 3 · h 1 = X 3 1 ( X 2 − 1 ) 2 . Whether there exist two univariate polynomials of X 2 : r 1 , r 2 such that f := q 1 · r 1 + q 2 · r 2 vanishes on H and has the desired leading term X 3 1 X 2 ? We find that as long as r 1 and r 2 satisfy g 1 h 1 LC ( g 3 ) · r 1 + LC ( h 2 ) · r 2 = 1 f := q 1 · r 1 + q 2 · r 2 is the vary polynomial we want. A new study on the vanishing ideal of a set of points with multiplicity