The vanishing ideal of a finite set of points with multiplicity structures
Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Grbner basis of the vanishing ideal under the lexicographic ordering. Our method discloses the essential geometric connection between the relative position of the points with multiplicity structures and the quotient basis of the vanishing ideal, so we will explicitly know the set of leading terms of elements of . We split the problem into several smaller ones which can be solved by induction over variables and then use our new algorithm for intersection of ideals to compute the result of the original problem. The new algorithm for intersection of ideals is mainly based on the Extended Euclidean Algorithm.
keywords:vanishing ideal, points with multiplicity structures, reduced Grbner basis, intersection of ideals.
To describe the problem, first we give the definitions below.
Definition 1: is called a lower set as long as if , lies in where with the 1 situated at the -th position (). For a lower set , we define its limiting set to be the set of all such that whenever , then .
As showed in Fig.1 below, there are three lower sets and their limiting sets. The elements of the lower sets are marked by solid circles and the elements of the limiting sets are marked by blank circles.
Fig.1: Illustration of three lower sets and their limiting sets.
Let be a field and be a point in the affine space , i.e. . Let be the polynomial ring over , where we write for brevity’s sake.
Definition 2: represents a point with multiplicity structure , where is a point in affine space and is a lower set. is called the multiplicity of point (here we use the definition in ). For each , we define a corresponding functional
Hence for any given finite set of points with multiplicity structures , we can define functionals where . Our aim is to find the reduced Grbner basis of the vanishing ideal under the lexicographic ordering with .
There exists an algorithm that provides a complete solution to this problem in . However, our answer for the special case of lexicographical ordering will be in a way more transparent than the one above. The ideas are summed-up as follows:
Construct the reduced Grbner basis of and get the quotient basis by induction over variables.
Get the quotient basis purely according to the geometric distribution of the points with multiplicity structures.
Split the original problem into smaller ones which can be converted into 1 dimension lower problems and hence can be solved by induction over variables.
Compute the intersection of the ideals of the smaller problems by using Extended Euclidean Algorithm.
There are several publications which have a strong connection to the work presented here. Paper  give a computationally efficient algorithm to get the quotient basis of the vanishing ideal over a set of points with no multiplicity structures and the authors introduce the lex game to describe the problem. Paper  offers a purely combinatorial algorithm to obtain the linear basis of the quotient algebra which can handle the set of points with multiplicity structures but it does not give the Grbner basis. For a finite set of points with multiplicity structures, our algorithm obtains a lower set by induction over variables and constructs the reduced Grbner bases at the same time. It is only by constructing Grbner basis we can prove that the lower set is the quotient basis.
One important feature of our method is the clear geometric interpretation, so in Section 2 an example together with some auxiliary pictures will be given in the first place to demonstrate this kind of feature which can make the algorithms and conclusions in this paper easier understood for us. In Section 3 and 4, some definitions and notions are given. Section 5 and 6 are devoted to our main algorithms of computing the reduced Grbner basis and the quotient basis together with the proofs. In Section 7 we demonstrate the algorithm to compute the intersection of two ideals and some applications.
First we give two different forms to represent the set of points with multiplicity structures.
For easier description, we introduce the matrix form which consists of two matrices with denoting the -th row vectors of and respectively. Each pair defines a functional in the following way.
And the functional set defined above is the same with that defined by in Section 1.
For example, given a set of three points with their multiplicity structures , where , the matrix form is like the follows.
For intuition’s sake, we also represent the points with multiplicity structures in a more intuitive way as showed in the left picture of Fig.2 where each lower set which represents the multiplicity structure of the corresponding point is also put in the affine space with the zero element (0,0) situated at . This intuitive representing form is the basis of the geometric interpretation of our algorithm.
We take the example above to show how our method works and what the geometric interpretation of our algorithm is like:
Step 1: Define mapping such that is mapped to . So consists of two -fibres: and as showed in the middle and the right pictures in Fig.2. Each fibre defines a new problem, so we split the original problem defined by into two small ones defined by and respectively.
Fig. 2: The left picture represents . The middle one is for and the right one for .
Step 2: Solve the small problems. Take the problem defined by for example.
First, it’s easy to write down one element of :
The geometry interpretation is: we draw two lines sharing the same equation of to cover all the points as illustrated in the left picture in Fig.3 and the corresponding polynomial is .
Fig. 3: Three ways to draw lines to cover the points.
According to the middle and the right pictures in Fig.2, we can write down another two polynomials in :
It can be checked that is the reduced Grbner basis of , and the quotient basis is . In the following, we don’t distinguish explicitly an -variable monomial with the element in , and we denote the quotient basis of by . Hence can be written as a subset of : , i.e. a lower set, denoted by .
In fact we can get the lower set in a more direct way by pushing the points with multiplicity structures leftward which is illustrated in the picture below (lower set is positioned in the right part of the picture with the (0,0) element situated at point (0,1)). The elements of the lower set in the right picture in Fig.4) are marked by solid circles. The blank circles constitute the limiting set and they are the leading terms of the reduced Grbner basis .
Fig.4: Push the points leftward to get a lower set.
In the same way, we can get the Grbner basis and a lower set for the problem defined by , where .
Step 3: Compute the intersection of the ideals and to get the result for the problem defined by .
First, we construct a new lower set based on in an intuitive way: let the solid circles fall down and the elements of rest on the elements of to form a new lower set which is showed in the right part of Fig.5 and the blank circles represent the elements of the limiting set .
Fig. 5: Get the lower set based on and .
Then we need to find polynomials vanishing on with leading terms being the elements of . Take for example to show the general way we do it.
We need two polynomials which vanish on and respectively, and their leading terms both have the same degrees of with that of the desired monomial and both have the minimal degrees of . It’s easy to notice that and satisfy the requirement and then we multiply and with respectively which are all univariate polynomials of to get two polynomials which both vanish on .
Next try to find two univariate polynomials of : such that vanishes on (which is apparently true already) and has the desired leading term .
To settle the leading term issue, write as univariate polynomials of . . Because and the highest degrees of of the leading terms of are both , we know that as long as the leading term of is , the leading term of is also .
Obviously if and only if we can keep the leading term of to be . In this case and will be just perfect. In our algorithm we use Extended Euclid Algorithm to compute .
Finally we obtain
which vanishes on and has as its leading term.
In the same way, we can get for , for and for . In fact we need to compute and in turn according to the lexicographic order because we need reduce by , reduce by and , and reduce by , and .
The reduced polynomial set can be proved in Section 6 to be the reduced Grbner basis of the intersection of two ideals which is exactly the vanishing ideal over , and is the quotient basis.
First, we define the following mappings.
Let , and naturally we define , and where . In fact we can apply these mappings to any set or any matrix of columns, because there is no danger of confusion. For example, let be a matrix of columns, and is a matrix of columns with the first columns of reserved and the last one eliminated.
The mapping embeds an dimensional lower set into the dimensional space. When the operation parameter is zero, we can get an dimensional lower set by mapping each element to as showed below.
Fig. 6: Embed the lower set in 2-D space into 3-D space with parameter .
Blank circles represent the elements of the limiting sets. Note that after the mapping, there is one more blank circle. In this case, the limiting set is always increased by one element .
In the case the operation parameter is not zero, it is obvious that what we got is not a lower set any more. But there is another intuitive fact we should realize.
Theorem 1: are dimensional lower sets, and . Let Then is an dimensional lower set, and where .
Proof: First to prove is a lower set. let , then i.e. . Because is a lower set, hence for if , then where with the 1 situated at the -th position. So . For , if , then we are finished. Else there must be . Because if , we have . Since we already have , this is contradictory to .
Second, , . If is a zero tuple, then must be , that is Else we know . If , then . Then , that is . Finally with the operation we have where . So .
4 Addition of lower sets
In this section, we define the addition of lower sets which is the same with that in , the following paragraph and Fig.7 are basically excerpted from that paper with a little modification of expression.
To get a visual impression of what the addition of lower sets dose, look at the example in Fig.7. What is depicted there can generalizes to arbitrary lower sets and in arbitrary dimension and can be described as follows. Draw a coordinate system of and insert . Place a translate of somewhere on the -axis. The translate has to be sufficiently far out, so that and the translate do not intersect. Then take the elements of the translate of and drop them down along the -axis until they lie on top of the elements of . The resulting lower set is denoted by .
Fig. 7: Addition of and .
Intuitively, we define algorithm AOL to realize the addition of lower sets.
Algorithm AOL: Given two dimensional lower sets , determine another lower set as the addition of , denoted by .
[step 1]: ;
[step 2]: If return . Else pick
[step 2.1]: If =, add the last coordinate of with . Go to [step 2.1]. Else , go to [step 2].
Given dimensional lower sets , the addition we defined satisfies:
is a lower set,
These are all the same with that in . And the proof can be referred to it.
As implied in the example of Section 2, when we want to get a polynomial with leading term showed in the right part of Fig.8, we need two polynomials with the leading terms which are not the elements of the lower sets and have the same degrees of as and the minimal degrees of as showed in the left part of Fig.8. In other words, , . It’s easy to understand that these equations hold for the addition of three or even more lower sets.
We use algorithm GLT to get the leading terms and from respectively.
Algorithm GLT: Given , and an dimensional lower set satisfying . Determine another which satisfies that , and , denoted by .
[step 1]: Initialize such as and .
[step 2]: if , return r, else , go to [step 2].
Definition 3: For any , view it as an element in and define to be the leading coefficient of which is an univariate polynomial of .
Algorithm GLP: is an dimensional lower set, and , the leading term of is , algorithm GLP returns a polynomial in the ideal whose leading term is Denoted by
[step 1:] .
[step 2:] Select is a factor of .
[step 3:] where is an element of whose leading term is .
Remark 1: in [step 3]. Since has the minimal degree of according to algorithm GLT, there exists no element which is a factor of satisfying . Hence monomial in the algorithm does not conclude the variable .
5 Associate a lower set to a set of points with multiplicity structures
For any given set of dimensional points with multiplicity structures, we can construct an dimensional lower set by induction.
Univariate case: , then the lower set is .
To pass from to (), we first solve a Special case.
Special case: is a set of dimensional points with multiplicity structures where all the points share the same coordinates. Write in matrix form as and all the entries in the last column of matrix have the same values. Classify the row vectors of to get according to the values of the entries in the last column of matrix and we guarantee the corresponding relationship between the row vectors of matrix and matrix holds in (). All the entries in the last column of are the same and the entries of the last column of stay the same too. Then eliminate the last columns of and to get which represents a set of dimensional points with multiplicity structures, by induction we get a lower set in dimensional space. Then we set
Next we deal with the General case.
General case: is a set of dimensional points with multiplicity structures. Split the set of points: . The points of are in the same -fibre, i.e. they have the same coordinates , ,and According to the Special case, for each , we can get a lower set , then we set
We now proof is a lower set although it is easy to understand as long as the geometric interpretation involves. Since it is obviously true for Univariate case, induction over dimension would be helpful for the proof.
Proof: Assume is a lower set for the dimensional situation and now we prove the conclusion for dimensional situation ().
First to prove of the Special case is a lower set.
We claim that represents an dimensional set of points with multiplicity structures (). For any , define Let So can be written in the form of . Apparently is an dimensional lower set and can be viewed as the multiplicity structure of the point . Hence is an dimensional set of points with multiplicity structures.
What’s else, we assert is a sub-matrix of and is a sub-matrix of Because of the corresponding relationship between the row vectors in and , we need only to prove is a sub-matrix of . If it is not true, there exists a row vector of which is not a row vector of . That is, there exists () such that is an element of the lower set , and is not included in any lower set (). However since and , must be included in . Hence our assertion is true.
Since is a sub-matrix of and is a sub-matrix of According to the assumption of induction and the way we construct , we have where are both lower sets. Based on the Theorem 1 in Section 3, is a lower set, and .
Then to prove of General case is a lower set. Since are lower sets, and the addition of lower sets is also a lower set according to Section 4, is obviously a lower set. The proof is finished.
6 Associate a set of polynomials to
For every lower set constructed during the induction procedure showed in the last section, we associate a set of polynomials to it.
We begin with the univariate case as we did in the last section.
P-univariate case: , and . The set of polynomials associated to is .
Apparently, of P-univariate case satisfies the following Assumption.
Assumption: For any given dimensional set of points with multiplicity structures, there are the following conclusions. For any , there exists a polynomial where such that
The leading term of under lexicographic ordering is .
The exponents of all lower terms of lies in .
vanishes on .
When we construct the set of polynomials , we should make sure the assumption always holds. Now let us consider the dimensional situation and still begin with the special case.
P-Special case: Given a set of points with multiplicity structures or in matrix form . All the given points have the same coordinates, i.e. the entries in the last column of are the same. We compute following the steps below.
[step 2]: , define as a sub-matrix of containing all the row vectors whose last coordinates equal to . Extract the corresponding row vectors of to form matrix , and the corresponding relationship between the row vectors in and holds for and .
[step 3]: , eliminate the last columns of and to get which represents a set of points in dimensional space with multiplicity structures. According to the induction assumption, we have the polynomial set , associated to the lower set ,.
[step 4]: Multiply every element of with to get .
[step 5]: Eliminate the polynomials in whose leading term is not included in to get .
Theorem 2: The got in P-Special case satisfies the Assumption.
Proof: According to the Section 5, , represents an dimensional set of points with multiplicity structures for And . is a lower set and .
For , we have . It is easy to check that it satisfies the first three terms of the Assumption.
For any other element of , . So let be the element in such that . We have vanishes on whose leading term is and the lower terms belong to . According to the algorithm .
First it is easy to check that the leading term of is since .
Second, the lower terms of are all in the set