Polynomial factorization and curve decomposition algorithms Cristina Bertone To cite this version: Cristina Bertone. Polynomial factorization and curve decomposition algorithms. Mathematics [math]. Université Nice Sophia Antipolis; Università degli studi di Torino, 2010. English. tel-00560802 HAL Id: tel-00560802 https://tel.archives-ouvertes.fr/tel-00560802 Submitted on 30 Jan 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÁ DEGLI STUDI DI TORINO FACOLTÁ DI SCIENZE MATEMATICHE FISICHE E NATURALI UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS FACULTÉ DES SCIENCES DE NICE THÈSE DE DOCTORAT pour l’obtention du titre de Dottore in Scienza e Alta Tecnologia, indirizzo Matematica Docteur en Sciences, specialité Mathématiques présentée par Cristina Bertone Polynomial factorization and curve decomposition algorithms Soutenue le 26 mars 2010 devant le jury composé de : M. Alberto Conte, Examinateur M. André Galligo, Directeur M. Grègoire Lecerf, Examinateur M. Bernard Mourrain, Examinateur Mme. Margherita Roggero, Directeur M. Carlo Traverso, Rapporteur Introduction Consider a set of polynomials fF1;:::;Fsg ⊆ K[X1;:::;Xn], K a field, and consider its set of zeros n V (F1;:::;Fs) = fP 2 K jFi(P ) = 0; i = 1; : : : ; sg: n The set of points V (F1;:::;Fs) is an algebraic set or a variety of K , K the algebraic closure of K. Actually, this set of points is the same as V (a), where a is the ideal in K[X1;:::;Xn] generated by the set of polynomials fF1;:::;Fsg. An algebraic set is said to be irreducible if it can not be expressed as the union of two proper algebraic subsets, and it is said to be reducible otherwise. If we consider a reducible algebraic set and we write it as union of irreducible algebraic subsets, these are the irreducible components (or simply the components). We will consider an algebraic curve (that is, an algebraic set of dimension 1) in the affine space C2 or C3 and our aim is to find its irreducible components, i.e. the polynomials defining them. In both affine spaces, the decomposition of a curve exists and is unique, in the sense that there is a finite number of irreducible components which are uniquely determined. We are interested in developing a practical method to construct this decomposition. In mathematics, we often deal with geometric problems in higher dimensional spaces by means of induction: that is, once the problem is solved in a “smaller” space, you can solve the problem also in a “bigger” space through the known properties of the small one. This is what we plan to do, we will bring back the problem of decomposing a curve in C3 to the problem of decomposing a curve in C2. Nevertheless, at a first sight, the situation in C3 is dramatically different from the situation in C2. C2 : a curve C is an algebraic set of dimension 1, so it is defined by a principal ideal: V (a) = C with a = (f(X; Y )) ⊆ C[X; Y ]: Then, C is irreducible if and only if f(X; Y ) is irreducible in C[X; Y ]. The problem of decomposing C in its irreducible components C1;:::; Cs is then equivalent to the problem of computing the factorization of f(X; Y ) in C[X; Y ]. iii C3 : a curve C is not defined by a single polynomial, we need 2 (or more). In this case, the decomposition of C in C1;:::; Cs is equivalent to the primary decomposition of the ideal a, such that V (a) = C . In the thesis we will study the decomposition of curves defined by rational polynomials. The problem of absolute factorization (and rational factorization) was studied by many authors, who proposed more and more performing algorithms (see [45], [46] for a survey on the history of the subject). On the contrary, the problem of computing the primary decomposition of an ideal a of C[X1;:::;Xn] was tackled with different techniques (see for instance [21] and the refe- rences therein) and the research in this field is in continuous progress (for example, the case of 0-dimensional ideals, see [23]). However, there is not yet an efficient strategy for the general case. Two of the most recent strategies for curves in C3 rely on the reduction to the case of a set of points on a plane and use numerical computations (see [29], [73]). In fact, it is computationally advantageous to solve the problem passing from an algebraic set of dimension 1 in C3 to an algebraic set of dimension 0 in C2. Indeed passing from a curve in C3 to a curve in C2 would require hard computations. Nevertheless, this is exactly what we will do: once studied an absolute factorization algo- rithm for bivariate polynomials, we will try to use the same techniques for a curve in C3 by projection on a generic plane; the technique of generic projection and absolute factorization is “classical”, but we will be able to speed up computations since our techniques are modular. In Chapter 1, we study the problem for n = 2 , so we deal with the computation of the absolute factorization of a rational polynomial. This problem was extensively studied in the recent years (see [15], [68], [72], [16] and the references therein), but we focus here on Trager-Traverso Algorithm (see [24], [44], [78] and Section 1.2 with Algorithm 1). On the one hand the idea is to follow the same strategy, on the other one to improve it: we will exploit modular computations in order to obtain faster and better results. Indeed using this tool, we are able to describe an absolute irreducibility test (Algorithm 2) and then develop modular techniques (Section 1.4) in order to have an absolute factorization algorithm (Algo- rithm 4). Its main aim is the same as Trager-Traverso’s one, but it gets sharper results (in the sense of the degree of the algebraic extension). The absolute irreducibility test is based on some properties of the Newton polytope of the polynomial. The absolute factorization algorithm constructs an algebraic extension of Q which contains the coefficients of the absolute factors and has minimal degree. This is ob- tained through a careful choice of a prime integer p which ensures that we have a primitive element of the algebraic extension needed in Qp, the field of p-adic numbers (Lemma 1.4.9). Relying on randomness, we can say that generally this choice of p gives a good reduction of iv f(X; Y ), i. e. the factorization of f(X; Y ) modulo p is a p-adic approximation of its absolute factorization. We then use Hensel lifting to obtain a more precise p-adic approximation of a primitive algebraic element and we finally construct a univariate polynomial defining the algebraic extension using the LLL algorithm. In the appendix (Chapter A and B) we summarize the main definitions and properties con- cerning the LLL algorithm and the field of p-adic numbers. In Chapter 2 we underline the main differences and similarities between the cases n = 2 and n = 3: to do this, we start with some basic definitions and properties about the primary decomposition of an ideal and the affine Hilbert function. We consider a complete intersec- tion ideal a and we define the rational, the algebraic and the conjugated components for a primary decomposition, in parallel with the absolute factorization of a polynomial. Using Hilbert function, we have also the definitions of degree and multiplicity of a primary com- ponent. After this, we exactly state the output of our decomposition algorithm for a curve in C3, which is the same one of other existing algorithms ([29], [73]). For each irreducible alge- braic component we would like to construct a separator polynomial (Definition 2.2.6): it is a polynomial defining an algebraic surface which contains a component but not the other ones. Furthermore, we point out the problem of having a bound on the degree of a separator polynomial; if we could have such a bound, we could use it inside a numerical algorithm of decomposition to avoid extra and unnecessary computations. In Chapter 3 we focus on bounds on the degree of a separator polynomial using algebraic geometry invariants and arguments existing in literature. We look for such a bound in three different ways: through a plane section, a particular case of the classical Lifting Problem in codimension 2 (Section 3.2); through the regularity of the ideal (Section 3.3) and through the generic initial ideal (Section 3.4). The Lifting Problem in codimension 2 is a classical one in algebraic geometry. Although the case of curves in the 3-dimensional space is completely solved, the problem is still open in higher dimensions. Laudal’s generalized trisecant lemma and the following improvements (see [50], [76], [38]) give a lower bound on the degree of a separator polynomial, by com- puting the generic plane section. About the Lifting Problem, we also briefly resume some original results concerning the positivity of the second Chern Class of a reflexive sheaf (Sec- tion 3.2.1), which is one of the directions of investigation to prove Mezzetti’s conjecture (Conjecture 3.2.1). The regularity is a well-known invariant for ideals, which bounds not only the degrees of the minimal generators of the ideal, but also their syzygies.