# Multiplicity Preserving Triangular Set Decomposition of Two Polynomials

###### Abstract

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by triangular sets, which may have negative multiplicities. In the bivariate case, we give a complete algorithm to decompose the system into multiplicity preserving triangular sets with positive multiplicities. We also analyze the complexity of the algorithm in the bivariate case. We implement our algorithm and show the effectiveness of the method with extensive experiments.

Keywords. Triangular set decomposition, multiplicity preserving decomposition, extended Euclidean algorithm.

## 1 Introduction

Decomposing a polynomial system into triangular sets is a classical method to solve polynomial systems. The method was first introduced by Ritt [20] and revised by Wu in his work of elementary geometry theorem proving [24, 25]. There exist many work about this topic [1, 3, 4, 6, 8, 10, 11, 13, 15, 16, 17, 18, 22, 27]. The main tool to decompose a polynomial system is pseudo-division. In most existing triangular decomposition methods based on the pseudo-division algorithm, one need to deal with the initial of certain polynomial(s), say , which will bring extraneous zeros. Usually, one decomposes here the system into two systems corresponding to the cases and . Doing so, the number of the systems increases quickly. Moreover, this leads to some repeated computations which can be avoided.

Another reason why we consider the topic is that the multiplicity of a component or a zero of a polynomial system is an important information which helps us to obtain a further understanding of the structure of the variety defined by the polynomial system.

Most triangular set decomposition algorithms do not preserve the multiplicities of the zeros or the components. One approach to remedy this drawback is to decompose the polynomial system into triangular sets first and then recover the multiplicities. Li proposed a method to compute the multiplicities of zeros of a zero-dimensional polynomial system after obtaining a triangular decomposition of the system [12]. Recently, Li, Xia, and Zhang proved that the characteristic sets in Wu’s sense for zero-dimensional polynomial system is actually multiplicity preserving with a minor modification [14]. They also gave a multiplicity preserving decomposition, but some of the components are not in triangular form.

In this paper, we use the concept of multiplicative variety, that is, the components and their multiplicities in the original polynomial system. We consider not only the components themselves but the multiplicities of these components. During the decomposition, the initials bring some extraneous multiplicative varieties in each pseudo-division step. We record them during the computation and remove them later, which helps us to recover the multiplicative varieties of the original system. We also avoid some repeated computation during the decomposition. Currently, the theory is complete for polynomial system with two polynomials. In particular, we provide a method to compute the multiplicative-zeros of a zero-dimensional bivariate system with two polynomials. We also analyze the complexity of the algorithm under some conditions.

Kalkbrener’s method for zero-dimensional bivariate polynomial system is similar to our method [17]. But his method is not multiplicity preserving. And our method is in a different sense: we remove the extraneous zeros from the system.

The paper is organized as below. In the next section, we provide some properties of primitive polynomial remainder sequences. In Section 3, we provide the theories to decompose a polynomial system with two polynomials into triangular sets which preserve the multiplicities of the components of the original system. We provide a multiplicity preserving algorithm to decompose a zero-dimensional bivariate polynomial system into triangular sets in Section 4. The complexity of the algorithm under some conditions are analyzed. Algorithms and examples are used to illustrate the effectiveness and efficiency of our method. We also compare our method with other related methods. We draw a conclusion in the last section.

## 2 Primitive Polynomial Remainder Sequence

In this section, we introduce some basic properties for primitive polynomial remainder sequences. In fact, there are many references for this topic, in particular [2, 16, 17]. We modify the procedure for our own purpose.

Let be a computable field with characteristic zero, such as the field of rational numbers and the polynomial ring in the indeterminates .

Let . We define

where means the coefficient of in and means the degree of in . is called primitive w.r.t. if .

The pseudo-division can be extended to the following form.

###### Lemma 2.1

Let , , , , and . There exist such that

(1) |

where is the leading coefficient of in , , . Furthermore, has the form:

(2) |

where . Moreover, if , then

(3) |

Proof. Write as univariate polynomials in ,

To eliminate the terms of with degree in , we have

where . It is clear that , since , and . So the lemma holds when . Note that when . Now, we need to eliminate from . If ,

where . Each term of contains a factor of the form . So , and . And , the results is still true. So the lemma holds when . Assuming that the lemma holds for the cases , then we have . If

Note that and the lowest power of in is larger than and . Then

We can similarly derive . When , we have . So has form (2) since the lowest power of is larger than . And since . So . since and . So . So the lemma holds for . The lemma is proved.

###### Corollary 2.2

Let be primitive, , and . Regard as univariate polynomials in . Then there exist an such that

and . Furthermore,

where represents the ideal generated by .

Proof. The corollary is obvious.

Proof. Regard as univariate polynomials in , and a polynomial in and , where . Let and . From Lemma 2.1, if . So . Since , . We have if . The lemma is proved.

The above result is a necessary condition to check whether has factors in .

###### Lemma 2.4

Let , . Assume that . Applying the extended Euclidean algorithm for w.r.t. the variable , we obtain a polynomial sequence , such that

(4) |

where , and , .

Proof. We prove the lemma by induction on . When , from Lemma 2.1, there exist , such that , where is the leading coefficient of in , . Let , and . It is clear that . Assume that for , (4) holds. Denote . For , we have , where . If is a factor of , set as the product of all the factors of in . Let . Then . Let

Remark: In most cases, we have and which helps us to design efficient algorithms.

###### Corollary 2.5

Let , and . From the extended Euclidean algorithm, we can obtain

(5) | |||||

(6) |

where , , , , and are primitive.

The following corollary is clear and useful.

## 3 Triangular Decomposition of Two Polynomials

In this section, we will give the method to decompose a system of two polynomials into triangular sets. We need the concept of multiplicity variety.

###### Definition 3.1

([9] pp. 129-130) Let be an unmixed variety of dimension in a projective space of dimension over . And

where is an irreducible variety of dimension and order . Let be the Chow form (see [9] pp.32) of ; which is irreducible over and of degree in , for . The form

(9) |

where are positive integers, satisfies the conditions for a Chow form of an algebraic variety which, regarded as a set of points, coincides with . We consider a new entity, consisting of the variety associated with the form , for a given choice of the exponents , denoted as . We write

(10) |

where corresponds to . We call a multiplicative variety and the multiplicity of . Especially, we call a multiplicative-zero set when .

Remark: Since an affine variety can be easily transformed into a projective variety, we will consider directly affine multiplicative varieties in in this paper. And we assume that the system has no solutions at .

###### Theorem 3.2

([9] pp160) are unmixed multiplicative varieties with dimension respectively. The intersection of and are with dimension . So are the intersections of and , and . Then we have

where represents the intersection of two multiplicative varieties, which preserves the multiplicities of each intersection component (for more details see [9], pp158-160).

###### Definition 3.3

([5] pp.139) Let be a zero dimensional ideal in such that the variety defined by consists of finitely many points in , where is algebraic closure of , and assume . Then the multiplicity of as a zero of , denoted by , is the dimension of the ring obtained by localizing at the maximal ideal corresponding to , that is:

###### Lemma 3.4

([5] pp. 144 ) Let be zero-dimensional, and have total degrees at most and no solutions at . If , where are independent variables, then there is a nonzero constant such that the Chow form of is

where and is the multiplicity of in .

The lemma illustrates the relationship between Chow form and the multiplicity of a point of a zero-dimensional polynomial system. The lemma tells us that is the multiplicity of the corresponding irreducible zero-dimensional component of the zero-dimensional polynomial system when in Definition 3.1.

From Theorem 3.2, we have the corollary below.

###### Corollary 3.5

Using the notations as Corollary 2.2, we have

The following lemma is important to our algorithm.

###### Lemma 3.6

Let be as Corollary 2.2. We have

(11) |

Proof. By Corollary 2.2, , so we have

(12) |

We can find that define a multiplicative variety with dimension since . Note that both are primitive and . And and both are dimensional.

We are going to prove that

(13) |

Assume that the Chow form of is as (9) and the order of is . From (3.5), we can assume that the Chow form of are and , respectively, where and . And the orders are and , respectively. We define the Chow form of as and the order as . Since the Chow form of the algebraic variety is not equal to zero, the definition is well defined. Here defines some components of , including the multiplicities of the components. Thus still has positive exponent for each simplified component. And we have

So (13) holds. Combining (12) and (13), we have (11). This ends the proof.

###### Definition 3.7

A multiplicity preserving triangular decomposition of a polynomial system is a group of triangular sets in multiplicative variety sense such that

(14) |

Remark: We will show that a multiplicity preserving triangular sets decomposition exists for systems with two polynomials in the rest of the paper. Note that for a zero-dimensional polynomial system, the existence of (14) is obvious. The existence of (14) for for general case (dimension mixed, more polynomials) is our future work.

The following is a key result of the paper.

###### Theorem 3.8

Proof. From (6), we have

Then by Corollary 3.5, for , we have

(16) | |||

(17) |

So we have

Remark: The decomposition is about the -dimensional component of .

###### Corollary 3.9

Use the notations as Corollary 2.6, we have

(18) | |||||

Proof. From Corollary 2.6, we have . So by (12) and Corollary 3.5, we have

(19) |

This helps us simplifying the computation. By (17), we have

(20) | |||||

From

we have

By (12) and Corollary 3.5, we have

And , so . Then we have (18).

Remark: The components , , and only involve polynomials in . Note that by Lemma 2.4, the coefficient of in for is zero when . These components can also be decomposed into triangular sets recursively. This corollary is very important since it provides a method to eliminate the main variable in ’s, which simplifies the decomposition. We can obtain another interesting phenomenon from simple observation, that is, the degree of all the resulting polynomials is bounded by the square of the degree of .

## 4 Multiplicity Preserving Decomposition for System of Two Bivariate Polynomials

In this section, we will consider the triangular decomposition of a zero-dimensional bivariate polynomial system with two polynomials, that is, . The method provided here is complete for a zero dimensional bivariate polynomial system with two polynomials. When is zero dimensional, defines a multiplicative-zero set.

### 4.1 Algorithm

###### Lemma 4.1

Using the similar notations as Corollary 2.6, if is zero-dimensional, we have