Daniel Lazard and Polynomial Systems A Personal View

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1 Daniel Lazard and Polynomial Systems A Personal View James H. Davenport Department of Computer Science University of Bath Bath BA2 7AY England J.H.Davenport@bath.ac.uk December 1, Introduction This paper was given at the International Conference on Polynomial System Solving, held in Paris November 2004, in honour of the official retirement of Professor Daniel Lazard. The views expressed here are purely those of the author, and do not pretend to be complete. 1.1 EUROSAM 1979 Luminy (Springer LNCS 72) In this, the first computer algebra conference to be formally published 1, I wish to draw attention to two particular articles. 1. B. Buchberger A criterion for detecting unnecessary reductions in the construction of Groebner bases, pp D. Lazard Systems of algebraic equations, pp (see also [18]) It is interesting to observe that these appeared in very different sessions: the polynomial one and the linear algebra one. 1.2 The state of the art in 1979 It is difficult today to realise how different our understanding and research in polynomial systems was 25 years ago. The author is grateful to all those who have tried to explain the theories of polynomial systems to him, but above all, of course, to Daniel Lazard himself. 1 As opposed to special issues of the SIGSAM Bulletin, or institutional technical reports. 1

2 Buchberger s article only considered the ordering that today we would call total degree reverse lexicographic. In fact, the proofs are valid, with no changes, for any admissible ordering, but the concept did not exist then. Lazard s article required a total degree article, but a strict reading of the article (I am sure this was not intended) would allow orders incompatible with multiplication. In general, one can say that the today s usual formalisation (for example [4]) largely did not exist, and certainly was not structured, 25 years ago. 2 The meeting of concepts In reality, we should not speak of a single meeting, because these meetings have been the work of many people over the years, and one could easily say that there is more to do in in this direction. As far as I am concerned, though, one of the turning points was the article Gröbner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations, which Lazard published in 1983 [22]. 2.1 The FGLM algorithm This algorithm [13] converts a zero-dimensional (finite number of solutions) Gröbner base with respect to one ordering < (typically total degree reverse lexicographic ) to another < (often purely lexicographic, so that one can use the Gianni Kalkbrener theorem [14, 15] to read off the solutions). It is certainly unnecessary to explain this algorithm to such an audience! However, I would rather make two observations about it. The enormous practical importance of this algorithm; The fact that this algorithm can easily be considered as a hybrid between the polynomial approach and the linear algebra approach. In brief, one enumerates the monomials in increasing order with respect to <, reducing each (a polynomial calculation) by the input basis with respect to <, and look for linear relations between these reducta, which are being viewed as a matrix of coefficients with the columns labelled by the relevant monomials. Such a linear relation, translated back to the original monomials enumerated wth respect to <, gives an element of the Gröbner base with respect to <. 3 Do we really want a single Gröbner Base? There may be occasions on which our real goal, say finding the solutions of a set of polynomial equations F, can be met by other techniques than computing a full Gröbner base of the ideal generated by F. 2

3 3.1 Factored Gröbner bases In the 1980 s, many people [12, 11] had the following idea. Suppose we are computing a Gröbner base, and currently G = {g 1,..., g n } defines a variety V, and g 1 (say) = h 1 h 1. Then, instead of continuing to apply Buchberger s algorithm to G, we can continue Buchberger s algorithm on two forks: G 1 = {h 1, g 2,..., g n } defining V 1, and G 2 = {h 1, g 2,..., g n } defining V 2, where V = V 1 V 2. Of course, this process can be done recursively. pro h i, h i will have lower degree than g 1: indeed it is common for them to reduce many of the other g i. Therefore it is normal for the computing time for G 1 and G 2 to total far less than the computing time for G itself. con If this happens, then all the gains from Buchberger s third criterion [8] and its generalisation [2] are lost one is essentially starting afresh. pro The structure of V 1 V 2 may be much easier to understand, particularly if V is not equi-dimensional. con We do not actually have a Gröbner base for V. con It is very hard to track multiplicities, and to avoid duplication. con Empirically, factorisation is rare except in a purely lexicographic order, which FGLM means we rarely use Buchberger s algorithm for directly. pro It is possible to devise fast probabilistic tests for irreducibility [12]. 3.2 Triangular sets We wish to describe the affine variety V (F ) (i.e. the common zeros) of a finite set of polynomials F. We will do this via a family {T 1,..., T r } of sets of polynomials. Let T 1 be a Gröbner basis of F. Let T i be such that distinct polynomials in T i have distinct main variables: we say that T i is triangular [30, but he called them characteristic sets ]. Say that ζ V (T i ) is regular [33] if it does not cancel the leading coefficient 2 of any member of T i. Then V (F ) is the union of the regular zeros of the T i, denoted by W (T i ). Unfortunately, there are several definitions of triangular sets, classified by [25] and summarised in [1]. The main ones are: characteristic sets [30, 33, though the definitions are slightly different [1]]; 2 With respect to whatever variable is in fact the leading variable of this particular polynomial. 3

4 regular chains and their representations [16]; normalized triangular sets [24]; tower of simple extensions [24]; regular sets [29]. characteristic sets The algorithm of [31] produces from F a family T 1,..., T l of characteristic sets of prime ideals such that V (F ) = l i=1 W (T i). [1, Theorem 3.3] gives some elegant properties of characteristic sets of prime ideals. Given F, the algorithm of [33] computes a characteristic set T of a finite set of polynomials G such that F and G generate the same ideal 3. 4 Is Lazard purely a specialist in polynomial systems? Certainly not. One could easily cite the following. His research on the factorisation of polynomials [20, 19]. His research on the algorithmics of Z[i] [21]. His work on the radicals of differential ideals [5]. His work on quantifier elimination, where in particular he showed that the purely algorithmic methods known at the time would give results far larger and clumsier than was necessary [23]. This and similar observations led to the development of partial cylindrical algebraic decomposition [9]. Later on he also found a better algorithm for the projection stage in cylindrical algebraic decomposition [26], and further developments of this are found in the latest software [7]. His improvement (with Rioboo) of the integration of rational functions, which avoided spurious singularities in the logarithmic part of the integral [27]. His algorithm (with Valibouze) for the inverse of the primitive element algorithm [28]. We give some notes on the importance of some of these points. 4.1 Cylindrical Algebraic Decomposition Cylindrical algebraic decomposition is an algebraic tool. However, we [6, 3] have been able to use it to solve analytic simplication problems currently a major 3 Note, however, that this does not mean that T is necessarily a characteristic set of F [1]. 4

5 challenge for all computer algebra systems. Consider for example the following pseudo-equality 4 : a b? = ab. The validity of this depends on a and b. For example, for 1 z 1 + z? = 1 z 2 the answer is positive, but for the apparently similar z 1 z + 1? = z 2 1, the answer is false if z = 2, the left-hand side is negative, but the righthand side is positive. 4.2 Integration analytic and algebraic Everyone (at least mathematicians would like to think so) knows that the indefinite integral of a function is determined up to a constant. Analytically, c is a constant iff x, y c(x) = c(y). Therefore, when we calculate a definite integral via an indefinite one, the constants cancel. Algebraically 5, a constant c is such that c = 0, so for example the Heaviside function is an algebraic constant. Hence it is important to avoid spurious singularities in an indefinite integral calculated algebraically [32, for example], otherwise the constant might not be an analytic constant. References [1] Aubry, P., Lazard, D., and Moreno Maza, M. On the Theories of Triangular Sets. J. Symbolic Comp. 28 (1999), [2] Backelin, J., and Fröberg, R. How we proved that there are exactly 924 cyclic 7-roots. In Proceedings ISSAC 1991 (1991), S. Watt, Ed., pp [3] Beaumont, J., Bradford, R., Davenport, J., and Phisanbut, N. A Poly-Algorithmic Approach to Simplifying Elementary Functions. In Proceedings ISSAC 2004 (2004), pp [4] Becker, T., and Weispfeninng, V. w. H. K. Groebner Bases. A Computational Approach to Commutative Algebra. Springer Verlag (1993). [5] Boulier, F., Lazard, D., Ollivier, F., and Petitot, M. Representation for the radical of a finitely generated differential ideal. In Proceedings ISSAC 1995 (1995), A. Levelt, Ed. [6] Bradford, R., and Davenport, J. Towards Better Simplification of Elementary Functions. In Proceedings ISSAC 2002 (2002), T. Mora, Ed., pp Similar issues apply to logarithms, and indeed all inherently multivalued inverse functions [10]. Futhermore, they are not limited to cases that involve complex numbers: see [6] for a discussion of arctan x + arctan y =? arctan 1 xy over R R. 5 x+y In the sense of Differential Algebra [17]. 5

6 [7] Brown, C. Improved Projection for Cylindrical Algebraic Decomposition. J. Symbolic Comp. 32 (2001), [8] Buchberger, B. A Criterion for Detecting Unnecessary Reductions in the Construction of Groebner Bases. In Proceedings EUROSAM 79 (1979), pp [9] Collins, G., and Hong, H. Partial Cylindrical Algebraic Decomposition for Quantifier Elimination. J. Symbolic Comp. 12 (1991), [10] Corless, R., Davenport, J., Jeffrey, D., and Watt, S. According to Abramowitz and Stegun. SIGSAM Bulletin 2 34 (2000), [11] Czapor, S. Solving Algebraic Equations: Combining Buchberger s Algorithm with Multivariate Factorization. J. Symbolic Comp. 7 (1989), [12] Davenport, J. Looking at a set of equations. Tech. Rep , [13] Faugère, J., Gianni, P., Lazard, D., and Mora, T. Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering. J. Symbolic Comp. 16 (1993), [14] Gianni, P. Properties of Gröbner bases under specializations. In Proceedings EUROCAL 87 (1989), pp [15] Kalkbrener, M. Solving systems of algebraic equations by using Gröbner bases. In Proceedings EUROCAL 87 (1989), pp [16] Kalkbrener, M. Three contributions to elimination theory. PhD thesis, Johannes Kepler University, [17] Kolchin, E. Differential Algebra and Algebraic Groups. Academic Press (1973). [18] Lazard, D. Résolution des Systèmes Comp. Sci. 15 (1981), d Équations Algébriques. Theor. [19] Lazard, D. Factorisation des Polynômes. Les Mathématiques de l Informatique (1982), [20] Lazard, D. On Polynomial Factorization. In Proceedings EUROCAM 82 [Springer Lecture Notes in Computer Science 144 (1982), pp [21] Lazard, D. On the Minimal Algorithm in Rings of Imaginary Quadratic Integers. J. Number Theory 15 (1982), [22] Lazard, D. Gröbner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations. In Proceedings EUROCAL 83 [Springer Lecture Notes in Computer Science 162 (1983), pp [23] Lazard, D. Quantifier Elimination: Optimal Solution for Two Classical Problems. J. Symbolic Comp. 5 (1988),

7 [24] Lazard, D. A New Method for Solving Algebraic Systems of Positive Dimension. Discr. Appl. Math. 33 (1991), [25] Lazard, D. Systems of algebraic equations (algorithms and complexity). Cortona Proceedings (1991). [26] Lazard, D. An Improved Projection Operator for Cylindrical Algebraic Decomposition. Algebraic Geometry and its Applications (1994). [27] Lazard, D., and Rioboo, R. Integration of Rational Functions - Rational Computation of the Logarithmic Part. J. Symbolic Comp. 9 (1990), [28] Lazard, D., and Valibouze, A. Computing Subfields: Reverse of the Primitive Element Problem. In Proceedings MEGA 92 (1993), F. Eysette and A. Galligo, Eds., pp [29] Moreno Maza, M. Calculs de Pgcd au-dessus des Tours d Extensions Simples et Résolution des Systèmes d Équations Algébriques. PhD thesis, Thesis, [30] Ritt, J. Differential Equations from an Algebraic Standpoint. Volume 14 (1932). [31] Ritt, J. Differential Algebra. American Mathematical Society Colloquium Proceedings vol. XXXIII (1950). [32] Trager, B. Algebraic Factoring and Rational Function Integration. In Proceedings SYMSAC 76 (1976), R. Jenks, Ed., pp [33] Wu, W.-T. A Zero Structure Theorem for Polynomial Equations Solving. MM Research Preprints 1 (1987),