# Triangular Decomposition of Semi-algebraic Systems

###### Abstract

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems.

We show that any such system can be decomposed into finitely many regular semi-algebraic systems. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t. the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.

6

ISSAC 2010,25–28 July 2010, Munich, Germany. \CopyrightYear2010 \crdata978-1-4503-0150-3/10/0007

I.1.2Symbolic and Algebraic ManipulationAlgorithms [Algebraic algorithms, Analysis of algorithms] \termsAlgorithms, Experimentation, Theory

## 1 Introduction

Regular chains, the output of triangular decompositions of systems of polynomial equations, enjoy remarkable properties. Size estimates play in their favor [12] and permit the design of modular [13] and fast [17] methods for computing triangular decompositions. These features stimulate the development of algorithms and software for solving polynomial systems via triangular decompositions.

For the fundamental case of semi-algebraic systems with rational number coefficients, to which this paper is devoted, we observe that several algorithms for studying the real solutions of such systems take advantage of the structure of a regular chain. Some are specialized to isolating the real solutions of systems with finitely many complex solutions [23, 10, 3]. Other algorithms deal with parametric polynomial systems via real root classification (RRC) [25] or with arbitrary systems via cylindrical algebraic decompositions (CAD) [9].

In this paper, we introduce the notion of a regular semi-algebraic system, which in broad terms is the “real” counterpart of the notion of a regular chain. Then we define two notions of a decomposition of a semi-algebraic system: one that we call lazy triangular decomposition, where the analysis of components of strictly smaller dimension is deferred, and one that we call full triangular decomposition where all cases are worked out. These decompositions are obtained by combining triangular decompositions of algebraic sets over the complex field with a special Quantifier Elimination (QE) method based on RRC techniques.

Regular semi-algebraic system. Let be a regular chain of for some ordering of the variables . Let and designate respectively the variables of that are free and algebraic w.r.t. . Let be finite such that each polynomial in is regular w.r.t. the saturated ideal of . Define . Let be a quantifier-free formula of involving only the variables of . We say that is a regular semi-algebraic system if: {itemizeshort}

defines a non-empty open semi-algebraic set in ,

the regular system specializes well at every point of (see Section 2 for this notion),

at each point of , the specialized system has at least one real zero. The zero set of , denoted by , is defined as the set of points such that is true and , , for all and all .

Triangular decomposition of a semi-algebraic system. In Section 3 we show that the zero set of any semi-algebraic system can be decomposed as a finite union (possibly empty) of zero sets of regular semi-algebraic systems. We call such a decomposition a full triangular decomposition (or simply triangular decomposition when clear from context) of , and denote by RealTriangularize an algorithm to compute it. The proof of our statement relies on triangular decompositions in the sense of Lazard (see Section 2 for this notion) for which it is not known whether or not they can be computed in singly exponential time w.r.t. the number of variables. Meanwhile, we are hoping to obtain an algorithm for decomposing semi-algebraic systems (certainly under some genericity assumptions) that would fit in that complexity class. Moreover, we observe that, in practice, full triangular decompositions are not always necessary and that providing information about the components of maximum dimension is often sufficient. These theoretical and practical motivations lead us to a weaker notion of a decomposition of a semi-algebraic system.

Lazy triangular decomposition of a semi-algebraic system. Let (see Section 3 for this notation) be a semi-algebraic system of and be its zero set. Denote by the dimension of the constructible set A finite set of regular semi-algebraic systems , is called a lazy triangular decomposition of if {itemizeshort}

holds, and

there exists such that the real-zero set contains and the complex-zero set either is empty or has dimension less than . We denote by LazyRealTriangularize an algorithm computing such a decomposition. In the implementation presented hereafter, LazyRealTriangularize outputs additional information in order to continue the computations and obtain a full triangular decomposition, if needed. This additional information appears in the form of unevaluated function calls, explaining the usage of the adjective lazy in this type of decompositions.

Complexity results for lazy triangular decomposition. In Section 4, we provide a running time estimate for computing a lazy triangular decomposition of the semi-algebraic system when has no inequations nor inequalities, (that is, when holds) and when generates a strongly equidimensional ideal of dimension . We show that one can compute such a decomposition in time singly exponential w.r.t. . Our estimates are not sharp and are just meant to reach a singly exponential bound. We rely on the work of J. Renagar [20] for QE. In Sections 5 and 6 we turn our attention to algorithms that are more suitable for implementation even though they rely on sub-algorithms with a doubly exponential running time w.r.t. .

A special case of quantifier elimination. By means of triangular decomposition of algebraic sets over , triangular decomposition of semi-algebraic systems (both full and lazy) reduces to a special case of QE. In Section 5, we implement this latter step via the concept of a fingerprint polynomial set, which is inspired by that of a discrimination polynomial set used for RRC in [25, 24].

Implementation and experimental results. In Section 6 we describe the algorithms that we have implemented for computing triangular decompositions (both full and lazy) of semi-algebraic systems. Our Maple code is written on top of the RegularChains library. We provide experimental data for two groups of well-known problems. In the first group, each input semi-algebraic system consists of equations only while the second group is a collection of QE problems. To illustrate the difficulty of our test problems, and only for this purpose, we provide timings obtained with other well-known polynomial system solvers which are based on algorithms whose running time estimates are comparable to ours. For this first group we use the Maple command Groebner:-Basis for computing lexicographical Gröbner bases. For the second group we use a general purpose QE software: qepcad b (in its non-interactive mode) [5]. Our experimental results show that our LazyRealTriangularize code can solve most of our test problems and that it can solve more problems than the package it is compared to.

We conclude this introduction by computing a triangular decomposition of a particular semi-algebraic system taken from [6]. Consider the following question: when does have a non-real root satisfying ? This problem can be expressed as , where and .

We call our LazyRealTriangularize command on the semi-algebraic system with the variable order . Its first step is to call the Triangularize command of the RegularChains library on the algebraic system . We obtain one squarefree regular chain , where and , satisfying . The second step of LazyRealTriangularize is to check whether the polynomials defining inequalities and inequations are regular w.r.t. the saturated ideal of , which is the case here. The third step is to compute the so called border polynomial set (see Section 2) which is with and . One can check that the regular system specializes well outside of the hypersurface . The fourth step is to compute the fingerprint polynomial set which yields the quantifier-free formula telling us that is a regular semi-algebraic system. After performing these four steps, (based on Algorithm 5, Section 6) the function call

LazyRealTriangularize |

in our implementation returns the following:

The above output shows that
forms a lazy triangular decomposition of the input semi-algebraic system.
Moreover, together with the output of the recursive calls,
one obtains a full triangular decomposition.
Note that the cases of the two recursive calls
correspond to and .
Since our LazyRealTriangularize uses the
Maple piecewise structure for formatting its output,
one simply needs to evaluate
the recursive calls with the `value`

command, yielding
the same result as directly calling RealTriangularize

where , , .

From this output, after some simplification, one could obtain the equivalent quantifier-free formula, , of the original QE problem.

##
2 Triangular decomposition of

Algebraic Sets

We review in the section the basic notions related to regular chains and triangular decompositions of algebraic sets. Throughout this paper, let be a field of characteristic 0 and be its algebraic closure. Let be the polynomial ring over and with ordered variables . Let be polynomials. Assume that . Then denote by mvar, init, and mdeg respectively the greatest variable appearing in (called the main variable of ), the leading coefficient of w.r.t. mvar (called the initial of ), and the degree of w.r.t. mvar (called the main degree of ); denote by der the derivative of w.r.t. mvar; denote by discrim the discriminant of w.r.t. mvar.

Triangular sets. Let be a triangular set, that is, a set of non-constant polynomials with pairwise distinct main variables. Denote by mvar the set of main variables of the polynomials in . A variable in is called algebraic w.r.t. if , otherwise it is said free w.r.t. . If no confusion is possible, we shall always denote by and respectively the free and the main variables of . Let be the product of the initials of the polynomials in . We denote by sat the saturated ideal of : if is the empty triangular set, then sat is defined as the trivial ideal , otherwise it is the ideal . The quasi-component of is defined as . Denote as the Zariski closure of .

Iterated resultant. Let and be two polynomials of . Assume is non-constant and let . We define as follows: if does not appear in , then ; otherwise is the resultant of and w.r.t. . Let be a triangular set of . We define by induction: if is empty, then ; otherwise let be the greatest variable appearing in , then , where and denote respectively the polynomials of with main variables equal to and less than .

Regular chain. A triangular set is a regular chain if: either is empty; or (letting be the polynomial in with maximum main variable), is a regular chain, and the initial of is regular w.r.t. sat. The empty regular chain is denoted by . Let . The pair is a regular system if each polynomial in is regular modulo sat. A regular chain or a regular system , is squarefree if for all , the der is regular w.r.t. sat.

Triangular decomposition. Let . Regular chains of form a triangular decomposition of if either: (Kalkbrener’s sense) or (Lazard’s sense). In this paper, we denote by Triangularize an algorithm, such as the one of [18], computing a triangular decomposition of the former kind.

Regularization. Let . Let be a regular chain of . Denote by Regularize an operation which computes a set of regular chains such that for each , either or is regular w.r.t. sat; we have , and for .

Good specialization [8]. Consider a squarefree regular system of . Recall that and stand respectively for mvar and . Let be a point of . We say that specializes well at if: none of the initials of the polynomials in vanishes modulo the ideal ; the image of modulo is a squarefree regular system.

Border polynomial [25]. Let be a squarefree regular system of . Let be the primitive and square free part of the product of all and all for and . We call the border polynomial of and denote by an algorithm to compute it. We call the set of irreducible factors of the border polynomial set of . Denote by an algorithm to compute it. Proposition 1 follows from the specialization property of subresultants and states a fundamental property of border polynomials.

###### Proposition 1

The system specializes well at if and only if the border polynomial .

##
3 Triangular decomposition of

semi-algebraic systems

In this section, we prove that any semi-algebraic system can be decomposed into finitely many regular semi-algebraic systems. This latter notion was defined in the introduction.

Semi-algebraic system. Consider four finite polynomial subsets , , , and of . Let denote the set of non-negative inequalities . Let denote the set of positive inequalities . Let denote the set of inequations . We will denote by the basic semi-algebraic system . We denote by the semi-algebraic system (SAS) which is the conjunction of the following conditions: , , and .

Notations on zero sets. In this paper, we use “” to denote the zero set of a polynomial system, involving equations and inequations, in and “” to denote the zero set of a semi-algebraic system in .

Pre-regular semi-algebraic system. Let be a squarefree regular system of . Let be the border polynomial of . Let be a polynomial set such that divides the product of polynomials in . We call the triple a pre-regular semi-algebraic system of . Its zero set, written as , is the set such that , , , for all , , . Lemma 1 and Lemma 2 are fundamental properties of pre-regular semi-algebraic systems.

###### Lemma 1

Let be a semi-algebraic system of . Then there exists finitely many pre-regular semi-algebraic systems , , s.t. .

The semi-algebraic system decomposes into basic semi-algebraic systems, by rewriting inequality of type as: . Let be one of those basic semi-algebraic systems. If is empty, then the triple , is a pre-regular semi-algebraic system. If is not empty, by Proposition 1 and the specifications of Triangularize and Regularize, one can compute finitely many squarefree regular systems such that holds and where is the border polynomial set of the regular system . Hence, we have where each is a pre-regular semi-algebraic system.

###### Lemma 2

Let be a pre-regular semi-algebraic system of . Let be the product of polynomials in . The complement of the hypersurface in consists of finitely many open cells of dimension . Let be one of them. Then for all , the number of real zeros of is the same.

###### Lemma 3

Let be a pre-regular semi-algebraic system of . One can decide whether its zero set is empty or not. If it is not empty, then one can compute a regular semi-algebraic system whose zero set in is the same as that of .

If , we can always test whether the zero set of is empty or not, for instance using CAD. If it is empty, we are done. Otherwise, defining , the triple is a regular semi-algebraic system. If is not empty, we solve the quantifier elimination problem and let be the resulting formula. If is false, we are done. Otherwise, by Lemma 2, above each connected component of , the number of real zeros of the system is constant. Then, the zero set defined by is the union of the connected components of above which possesses at least one solution. Thus, defines a nonempty open set of and is a regular semi-algebraic system.

###### Theorem 1

Let be a semi-algebraic system of . Then one can compute a (full) triangular decomposition of , that is, as defined in the introduction, finitely many regular semi-algebraic systems such that the union of their zero sets is the zero set of .

## 4 Complexity Results

We start this section by stating complexity estimates for basic operations on multivariate polynomials.

Complexity of basic polynomial operations. Let be polynomials with respective total degrees , and let . Let and be the height (that is, the bit size of the maximum absolute value of the numerator or denominator of a coefficient) of , the product and the resultant , respectively. In [14], it is proved that can be computed within bit operations where and . It is easy to establish that and are respectively upper bounded by and . Finally, let be a matrix over . Let (resp. ) be the maximum total degree (resp. height) of a polynomial coefficient of . Then can be computed within bit operations, see [15].

We turn now to the main subject of this section, that is, complexity estimates for a lazy triangular decomposition of a polynomial system under some genericity assumptions. Let . A lazy triangular decomposition (as defined in the Introduction) of the semi-algebraic system , which only involves equations, is obtained by the above algorithm.

Proof of Algorithm 1. The termination of the algorithm is obvious. Let us prove its correctness. Let , for be the output of Algorithm 1 and let for be the regular chains such that . By Lemma 3, each is a regular semi-algebraic system. For , define . Then we have , where each is equidimensional. For each , by Proposition 1, we have

Moreover, we have

Hence,

holds. In addition, since is regular modulo , we have

and . So the , for , form a lazy triangular decomposition of .

In this section, under some genericity assumptions for , we establish running time estimates for Algorithm 1, see Proposition 3. This is achieved through: {itemizeshort}

Proposition 2 giving running time and output size estimates for a Kalkbrener triangular decomposition of an algebraic set, and

Theorem 2 giving running time and output size estimates for a border polynomial computation. Our assumptions for these results are the following: {itemizeshort}

is equidimensional of dimension ,

are algebraically independent modulo each associated prime ideal of the ideal generated by in ,

consists of polynomials, . Hypotheses and are equivalent to the existence of regular chains of such that are free w.r.t. each of and such that we have

Denote by , respectively the maximum total degree and height of . In her PhD Thesis [22], Á. Szántó describes an algorithm which computes a Kalkbrener triangular decomposition, , of . Under Hypotheses to , this algorithm runs in time counting operations in , while the total degrees of the polynomials in the output are bounded by . In addition, are square free, strongly normalized [18] and reduced [1].

From , we obtain regular chains forming another Kalkbrener triangular decomposition of , as follows. Let and . Let be the polynomial of with as main variable. Let be the primitive part of regarded as a polynomial in . Define . According to the complexity results for polynomial operations stated at the beginning of this section, this transformation can be done within operations in .

Dividing by its initial we obtain a monic polynomial of . Denote by the regular chain . Observe that is the reduced lexicographic Gröbner basis of the radical ideal it generates in . So Theorem 1 in [12] applies to each regular chain . For each polynomial , this theorem provides height and total degree estimates expressed as functions of the degree [7] and the height [19, 16] of the algebraic set . Note that the degree and height of are upper bounded by those of . Write where each is a monomial and are in such that holds. Let be the lcm of the ’s. Then for and each : {itemizeshort}

the total degree is bounded by and,

the height by . Multiplying by brings back. We deduce the height and total degree estimates for each below.

###### Proposition 2

The Kalkbrener triangular decomposition of can be computed in operations in . In addition, every polynomial has total degree upper bounded by , and has height upper bounded by .

Next we estimate the running time and output size for computing the border polynomial of a regular system.

###### Theorem 2

Let be a squarefree regular system of , with and . Let be the border polynomial of . Denote by , respectively the maximum total degree and height of a polynomial in . Then the total degree of is upper bounded by , and can be computed within bit operations.

Define . We need to compute the iterated resultants , for all . Let . Observe that the total degree and height of are bounded by and respectively. Define , …, , …, . Let . Denote by and the total degree and height of , respectively. Using the complexity estimates stated at the beginning of this section, we have and . Therefore, we have . From these size estimates, one can deduce that each resultant (thus the iterated resultants) can be computed within bit operations, by the complexity of computing a determinant stated at the beginning of this section.

Hence, the product of all iterated resultants has total degree and height bounded by and , respectively. Thus, the primitive and squarefree part of this product can be computed within bit operations, based on the complexity of a polynomial gcd computation stated at the beginning of this section.

###### Proposition 3

From the Kalkbrener triangular decomposition of Proposition 2, a lazy triangular decomposition of can be computed in bit operations. Thus, a lazy triangular decomposition of this system is computed from the input polynomials in singly exponential time w.r.t. , counting operations in .

For each , let be the border polynomial of and let (resp. ) be the height (resp. the total degree) bound of the polynomials in the pre-regular semi-algebraic system . According to Algorithm 1, the remaining task is to solve the QE problem for each , which can be solved within bit operations, based on the results of [20]. The conclusion follows from the size estimates in Proposition 2 and Theorem 2.

## 5 Quantifier Elimination by RRC

In the last two sections, we saw that in order to compute a triangular decomposition of a semi-algebraic system, a key step is to solve the following quantifier elimination problem:

(1) |

where is a pre-regular semi-algebraic system of . This problem is an instance of the so-called real root classification (RRC) [27]. In this section, we show how to solve this problem when is what we call a fingerprint polynomial set.

Fingerprint polynomial set. Let be a pre-regular semi-algebraic system of . Let . Let be the product of all polynomials in . We call a fingerprint polynomial set (FPS) of if: {itemizeshort}

for all , for all we have:

,

for all with and , , if the signs of and are the same for all , then has real solutions if and only if does.

Hereafter, we present a method to construct an FPS based on projection operators of CAD.

Open projection operator [21, 4]. Hereafter in this section, let be ordered variables. Let be non-constant. Denote by factor the set of the non-constant irreducible factors of . For , define . Let (resp. ) be the set of the polynomials in factor with main variable equal to (resp. less than) . The open projection operator () w.r.t. variable maps to a set of polynomials of defined below:

Then, we define .

Augmentation. Let and . Denote by der the derivative closure of w.r.t. , that is, The open augmented projected factors of is denoted by and defined as follows. Let be the smallest positive integer such that holds. Denote by the set factor; we have {itemizeshort}

if , then ;

if , then

###### Theorem 3

Let be finite and let be a map from to the set of signs . Then the set is either empty or a connected open set in .

By induction on . When , the conclusion follows from Thom’s Lemma [2]. Assume . If is not the smallest positive integer such that holds, then can be written and the conclusion follows by induction. Otherwise, write as , where and . We have: . Denote by the set . If is empty then so is and the conclusion is clear. From now on assume not empty. Then, by induction hypothesis, is a connected open set in . By the definition of the operator , the product of the polynomials in is delineable over w.r.t. . Moreover, is derivative closed (may be empty) w.r.t. . Therefore is either empty or a connected open set by Thom’s Lemma.

###### Theorem 4

Let be a pre-regular semi-algebraic system of . The polynomial set is a fingerprint polynomial set of .

Recall that the border polynomial of divides the product of the polynomials in . We have . So satisfies in the definition of FPS. Let us prove . Let be the product of the polynomials in . Let such that both , hold and the signs of and are equal for all . Then, by Theorem 3, and belong to the same connected component of , and thus to the same connected component of . Therefore the number of real solutions of and that of are the same by Lemma 2.

From now on, let us assume that the set in the pre-regular semi-algebraic system is an FPS of . We solve the quantifier elimination problem (1) in three steps: compute at least one sample point in each connected component of the semi-algebraic set defined by ; for each sample point such that the specialized system possesses real solutions, compute the sign of for each ; generate the corresponding quantifier-free formulas.

In practice, when the set is not an FPS, one adds some polynomials from , using a heuristic procedure (for instance one by one) until Property