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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
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
A new study on the vanishing ideal of a set of points with multiplicity
A new study on the vanishing ideal of a set of points with multiplicity
D ⊆ Nn
0 is called a lower set as long as whenever
d = (d1, . . . , dn) lies in D and di = 0 , d − ei also lies in D where ei = (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 ∈ Nn
0 − D such that whenever βi = 0,
then β − ei ∈ 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
p, D represents a point p with multiplicity structure D, where p is a point in affine space An and D is a lower set. ♯D is called the multiplicity of point p. H = {p1, D1, . . . , pt, Dt} is a set of points with multiplicity structures.
A new study on the vanishing ideal of a set of points with multiplicity
Given p, D, for each d = (d1, . . . , dn) ∈ D, we define a corresponding functional L(f) = ∂d1+...+dn ∂xd1
1 . . . ∂xdn n
f(p). For H = {p1, D1, . . . , pt, Dt}, we can define m functionals where m ♯D1 + . . . + ♯Dt. Our aim is to find the reduced Gr¨
vanishing ideal I(H) = {f ∈ k[X]; Li(f) = 0, i = 1, . . . , m} under the lexicographic ordering with X1 ≻ X2 ≻ . . . ≻ Xn.
A new study on the vanishing ideal of a set of points with multiplicity
[1]M.G. Marinari, H.M. M-
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
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
into the reduced Gr¨
understand the problem better and obtain some new conclusions.
A new study on the vanishing ideal of a set of points with multiplicity
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 xn of the points. In each subset the points have the common value of xn. Compute the reduced Gr¨
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 xn of the points. In each subset the points have the common value of xn. Compute the reduced Gr¨
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 xn of the points. In each subset the points have the common value of xn. Compute the reduced Gr¨
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
A new study on the vanishing ideal of a set of points with multiplicity
Two representing forms of points with multiplicity structures Given H = {p1, D1, p2, D2, p3, D3}, where p1 = (1, 1), D1 = {(0, 0), (0, 1), (1, 0)} p2 = (2, 1), D2 = {(0, 0), (0, 1), (1, 0), (1, 1)} p3 = (0, 2), D3 = {(0, 0), (1, 0)}.
A new study on the vanishing ideal of a set of points with multiplicity
P = 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 2 , D = 1 1 1 1 1 1 1 .
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 Xn coordinates) Given H = {p1, D1, p2, D2}, where p1 = (1, 1), D1 = {(0, 0), (0, 1), (1, 0)}, p2 = (2, 1), D2 = {(0, 0), (0, 1), (1, 0), (1, 1)}. All the points share the same X2 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.
P1 = 1 1 1 1 2 1 2 1 , D1 = 1 1 .
P2 = 1 1 2 1 2 1 , D2 = 1 1 1 1 .
A new study on the vanishing ideal of a set of points with multiplicity
matrix P1, and b for that of D1, that is a := 1, b := 0. And then eliminate the last columns of matrices P1 and D1 to get an univariate problem which can be solved according to the induction assumption. Quotient basis: ˆ D1 := {1, X1, X 2
1 , X 3 1 }.
Gr¨
G1 := {(X1 − 1)2(X1 − 2)2}. ˜ D1 := embedb(ˆ D1),˜ G1 := mapga,b( ˆ G1). embedc : D
′ ∈ Nn−1
− → D
′′ ∈ Nn
0,
(d1, . . . , dn−1) − → (d1, . . . , dn−1, c). mapga,b : G
′ (n − 1-variable polynomial set) −
→ G
′′
(n-variable polynomial set) g − → g · (Xn − a)b.
A new study on the vanishing ideal of a set of points with multiplicity
D2 and ˜ G2.
D := ˜ D1 ˜ D2 is the quotient basis. G := ˜ G1 ˜ G2
is the reduced Gr¨
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 = {p1, D1, p2, D2, p3, D3}, where p1 = (1, 1), D1 = {(0, 0), (0, 1), (1, 0)}, p2 = (2, 1), D2 = {(0, 0), (0, 1), (1, 0), (1, 1)}, p3 = (0, 2), D3 = {(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 X2 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 h1 = (X2 − 1)2; g1 = X2 − 2; h2 = (X2 − 1)(X1 − 1)(X1 − 2)2; g2 = X 2
1 ;
h3 = (X1 − 1)2(X1 − 2)2. f1 := g1 · h1.
A new study on the vanishing ideal of a set of points with multiplicity
1 X2:
g3 := X1 · g2, the leading terms of g3 and h2 share the same degree of X1 with X 3
1 X2 as a factor.
q1 := h2 · g1 = (X2 − 1)(X1 − 1)(X1 − 2)2(X2 − 2), q2 := g3 · h1 = X 3
1 (X2 − 1)2.
Whether there exist two univariate polynomials of X2: r1, r2 such that f := q1 · r1 + q2 · r2 vanishes on H and has the desired leading term X 3
1 X2?
We find that as long as r1 and r2 satisfy g1 LC(g3) · r1 + h1 LC(h2) · r2 = 1 f := q1 · r1 + q2 · r2 is the vary polynomial we want.
A new study on the vanishing ideal of a set of points with multiplicity
Lemma 1: G is the reduced Gr¨
n-variable polynomial ideal under lexicographic ordering. Define p0(G) as the univariate polynomial of Xn in G. View all the elements of G as polynomials of K(Xn)[X1, . . . , Xn−1]. For any element g ∈ G, define LC(g) to be the leading coefficient
conclusion that LC(g) is always a factor of p0(G).
A new study on the vanishing ideal of a set of points with multiplicity
A new study on the vanishing ideal of a set of points with multiplicity
We don’t distinguish an element in Nn
0 with an n-variable
monomial and hence can view a lower set as a set of
Gr¨
Definitions: proj : D − → k (d1, . . . , dn) − → dn.
′ ∈ Nn
0 −
→ D
′′ ∈ Nn−1
(d1, . . . , dn) − → (d1, . . . , dn−1). embedc : D
′ ∈ Nn−1
− → D
′′ ∈ Nn
(d1, . . . , dn−1) − → (d1, . . . , dn−1, c).
A new study on the vanishing ideal of a set of points with multiplicity
Addition of lower sets: Given two lower sets D1, D2 ⊂ Nn
0, determine another lower set as the addition of D1
and D2. Denoted by D := D1 + D2. [step 1]: D := D1; [step 2]: If ♯D2 = 0 return D. Else pick a ∈ D2, D2 := D2 \ {a}. [step 2.1]: If a ∈ D, add the last coordinate of a with 1. Go to [step 2.1]. Else D := D {a}, go to [step 2].
A new study on the vanishing ideal of a set of points with multiplicity
Algorithm GLT: Given a ∈ Nn
0 and a lower set D ⊂ Nn
satisfying a / ∈ D. Determine r = (r1, . . . , rn) ∈ Nn
0 satisfying that
r / ∈ D, proj(r) = proj(a) and (r1, . . . , rn−1, rn − 1) ∈ D, denoted by r := GLT(a, D). [step 1]: Initialize r such as proj(r) = proj(a) and proj(r) = 0. [step 2]: if r / ∈ D, return r, else rn := rn + 1, go to [step 2].
A new study on the vanishing ideal of a set of points with multiplicity
Algorithm GLP: G is a reduced Gr¨
and a / ∈ Q(G). This algorithm returns a polynomial p ∈ G with the leading term GLT(a, Q(G)). Denoted by p := GLP(a, G). [step 1:] c := GLT(a, Q(G)). [step 2:] Select c
′ ∈ E(Q(G)), s.t. c ′ is a factor of
c′ .
[step 3:] p := fc′ · d where fc′ is an element of G whose leading term is c
′. A new study on the vanishing ideal of a set of points with multiplicity
G1 and G2 are the reduced Gr¨
ideals under lexicographic ordering. Algorithm Intersection computes the intersection of these two ideals as long as GCD(p0(G1), p0(G2)) = 1. Algorithm Intersection: G1 and G2 are the reduced Gr¨
GCD(p0(G1), p0(G2)) = 1. Return the reduced Gr¨
G:=Intersection(G1,G2).
A new study on the vanishing ideal of a set of points with multiplicity
[Step 1:] D := Q(G1) + Q(G2). View E(D) as a monomial
[Step 2:] If E(D) = ∅, the algorithm is done. Else select the minimum element of E(D), denoted by T. E(D) := E(D)/{T}. [Step 3:] f1 := GLP(T, G1), f2 := GLP(T, G2). q1 := f1 · p2, q2 := f2 · p1. [Step 4:] t1 :=
p2 LC(f2), t2 := p1 LC(f1).
[Step 5:] Use Extended Euclidean Algorithm to find r1, r2 s.t. r1 · t1 + r2 · t2 = 1. [Step 6:] f := q1 · r1 + q2 · r2. Reduce f with G to get f
′,
and G := G {f
′}. Go to [Step 2]. A new study on the vanishing ideal of a set of points with multiplicity
For arbitrary set of points with arbitrary multiplicity structures in arbitrary dimensional affine space, our algorithm can compute the reduced Gr¨
variables.
A new study on the vanishing ideal of a set of points with multiplicity
Univariate case: Given H = {p1, D1, . . . , pt, Dt}, the quotient basis is D(H) = {0, 1, . . . ,
t
♯Di} and the reduced Gr¨
I(H) = {
t
(X1 − pi)♯Di}.
A new study on the vanishing ideal of a set of points with multiplicity
Assume the (n − 1)-variable (n ≥ 2) problem has been
A new study on the vanishing ideal of a set of points with multiplicity
p-Special case: Given a set of n dimensional points with multiplicity structures H = {p1, D1, . . . , pt, Dt} or in matrix form P = (pij)m×n, D = (dij)m×n. All the given points have the same Xn coordinates, i.e. the elements of the last column of P are the same. We get the reduced Gr¨
quotient basis D by following the steps below. [step 1]: c := p1n; w := max{din; i = 1, . . . , m}; [step 2]: ∀i = 0, . . . , w, define SDi as a sub-matrix of D containing all the row vectors whose last coordinates are i. Extract the corresponding row vectors of P to form matrix SPi. [step 3]: ∀i = 0, . . . , w, eliminate the last column of SPi and the last column of SDi to get ˜ SPi and ˜ SDi which both have n − 1 columns. According to the assumption, we have the corresponding lower set ˜ Di and ˜ Gi = I( ˜ SPi, ˜ SDi) for short.
A new study on the vanishing ideal of a set of points with multiplicity
[step 4]: D := w
i=0 embedi(˜
Di). Multiply every element of ˜ Gi with (Xn − c)i to get Gi. ˜ G := w
i=0 Gi
{(Xn − c)w+1}. [step 5]: Eliminate the polynomials in ˜ G whose leading terms are not in E(D) to get G. Return {G, D}.
A new study on the vanishing ideal of a set of points with multiplicity
Main algorithm MainG: Given a set of points with multiplicity structures H or in matrix form P = (pij)m×n, D = (dij)m×n, we get the reduced Gr¨
quotient basis D, denoted by {G,D}:=MainG(H). [step 1]: Write H as H = H1 H2 . . . Hs where Hi(1 ≤ i ≤ s) are sets of points with multiplicity structures, all the points in Hi have the same Xn coordinates ci, i = 1, . . . , s, where ci = cj, ∀i, j = 1, . . . , s, i = j. [step 2]: According to p-Special case, we have D
′
i = D(H), Gi = poly(Hi).
[step 3]: D := D
′
1, G := G1, i := 2.
A new study on the vanishing ideal of a set of points with multiplicity
[step 4]: If i > s, go to [step 5]. [step 4.1]: D := D + D
′
i ; ˆ
G := ∅. View E(D) as a monomial set MS := E(D). [step 4.2]: If ♯MS = 0, go to [step 4.7], else select the minimal element LT of MS under lexicographic ordering, MS := MS \ {LT}. [step 4.3]: f1 := GLP(LT, G), f2 := GLP(LT, Gi). q1 := f1 · p0(Gi), q2 := f2 · p0(G).
A new study on the vanishing ideal of a set of points with multiplicity
[step 4.4]: t1 := p0(Gi) LC(f2), t2 := p0(G) LC(f1). [step 4.5]: Use Extend Euclidean Algorithm to compute r1 and r2 s.t. r1 · t1 + r2 · t2 = 1. [step 4.6]: f := r1 · q1 + r2 · q2. Reduce f with the elements in ˆ G to get f
′; ˆ
G := ˆ G {f
′}. Go to [step 4.2].
[step 4.7]: G := ˆ
[step 5]: Return {G, D}.
A new study on the vanishing ideal of a set of points with multiplicity
Theorem:
Given a set of points with multiplicity structures H, the algorithm MainG returns {G, D}, where G is the reduced Gr¨
lexicographic ordering, and D is the corresponding quotient basis.
A new study on the vanishing ideal of a set of points with multiplicity
A new study on the vanishing ideal of a set of points with multiplicity