SALSA Software for ALgebraic Systems Ro cq uencou rt Un it and Applications http://fgbrs.lip6.fr/salsa Proposed in November 2004 - Created in January 2006 Evaluation period : 2002 - 2004 (SPACES Project) - 2004 - 2006 (SALSA Project)
The SALSA project Joint team with UPMC (Université Pierre et Marie Curie, Paris 6) and LIP6 (Laboratoire d’Informatique de Paris 6). Permanent (4) : • Fabrice Rouillier - DR - INRIA • Jean-Charles Faugère - CR - CNRS • Mohab Safey El Din - Ass. Professor - UPMC • Philippe Trébuchet - Ass. Professor - UPMC External/Misc. (2) : • Daniel Lazard - Retired professor - UPMC • Magali Bardet - Ass. Professor - Université de Caen. PhD students (5) : Sylvain Lacharte - Guillaume Moroz - Rahmany Sajjad - Rong Xiao - Liang Ye.
� , p r } , F = { f 1 , � , f s } , with p i , f i ∈ K [ X 1 , � , X n ] � , p r = 0 , f 1 � 0 , � , f s � 0 } General Objectives � , p r = 0 , f 1 > 0 , � , f s > 0 } E = { p 1 , ¯ n , p 1 = 0 , C = { x ∈ K If K = Q , S = { x ∈ R n , p 1 = 0 , “Solve” C (or S ) when K = Q , Z / p Z or F q Computable objects - Algorithms - Implementations - Applica- tions - Complexity. SALSA’s specificities : • Only certified or exact results; • Always at least one solution for the general case; • Checkable assumptions for particular cases; • Full chain of algorithms for solving end-user problems;
General Objectives (1) A natural split (with the end-user as well as with the math. point of view) : • Zero-dimensional systems : ◦ count the real / complex roots ; ◦ detect multiple points and compute multiplicities ; ◦ provide an accurate and certified approximation of the roots; ◦ signs of polynomials at the real roots of a system ; • Parametric systems : ◦ count the real / complex roots wrt parameter’s values ; ◦ describe geometrically the solutions set ; ◦ provide formal expressions of the roots ; ◦ check/compute numerically stable solutions.
General Objectives (2) • General systems : ◦ Testing (real) emptiness; ◦ Computing one point per semi-algebraically connected compo- nent (black box for parametric systems and for quantifier elimination). • Quantifier elimination : (long term project ...). Use of algorithms for general semi-algebraic sets and/or parametric systems within classical simplification processes (Tarski principle).
Main Applications Our priorities are dictated by selected longstanding studies : • Cryptography/Coding - Gröbner engine without coefficient growth constraints; Collaboration with the CODES project - New field : “Algebraic Cryptanalysis” - HFE Challenge - conferences XXCrypt - Contracts with CELAR (DGA) - Thales. • Signal processing : ◦ Hyperfrequencies filters - Collaboration with APICS (INRIA Sophia) - Real roots of zero-dimensional systems with a large number of variables. ARC SILA - Contribution to a dedicated software (Dedale) - International Journal of RF & Computer-Aided Engeneering. ◦ Optimization problems - Collaboration with Deaking univer- sity : real roots of parametric systems. Qualitative study of the convergence of numerical optimization methods for wire- less networks - IEEE transactions on signal processing.
Main Applications (2) • Robotics : ◦ Collaboration with COPRIN project and IRCCyN team (CNRS laboratory - Nantes). Real roots of zero-dimen- sional/parametric systems - Direct kinematics problem - cusp- idal manipulators. AS CNRS 2002-2005 for serial cuspidal robots. ANR project SIROPA in 2007 for parallel robots. • Computational Geometry : ◦ Intersections of quadrics - Collaboration with VEGAS team. Structured “small” zero-dimensional systems (see VEGAS’ pre- sentation). ◦ Study of ridges - Collaboration with GEOMETRICA and now VEGAS. Zero-dimensional systems with few variables but a large number of solutions - topology of curves. Dedicated soft- ware - J. of CAGD
SALSA Software SALSA Software is made of : • FGb/Gb : Gröbner bases computations (J.-C. Faugère) - C/C++ • RS : Real roots of zero-dimensional systems (F. Rouillier) - C • DV : Parametric systems (G. Moroz, F. Rouillier) - Maple lib. • RAGlib : critical point methods and roadmaps (M. Safey El Din) - Maple lib • Interface between Maple and RS/Gb/FGb Experimental package : macrev (P. Trébuchet) - Generalized normal forms - C/C++ Development packages : GC (C - Memory management), MPFI (multi- precision interval arithmetic), MPAI (infinitesimal arithmetic). FGb and RS will be part of Maple 11 (dynamically linked)
� , p r = 0 , f 1 � 0 , � , f s � 0 } p j , f i ∈ Q [ U 1 , � , U d , X d +1 , � , X n ] Results - algebraic preprocessing { p 1 = 0 , Modelization / First informations (degree, dimension, shape of some solutions, elimination of variables, saturation of ideals, etc.) • Basic algorithm F 4 (1999 - Faugère) - Gröbner bases • Main improvement F 5 (2002 - Faugère) - Gröbner bases • Under study ◦ F 7 - Gröbner bases (Faugère) - decompositions into primes ◦ Generalized normal forms in positive dimension (P. Trébuchet) After this step : kind of problem to be solved - zero-dimensional - para- metric or positive dimensional. Eventually : change the modelization (new end-user query - specific prop- erties, etc.). For example : saturate wrt to f i , remove components without physical meanings.
� , p r = 0 , f 1 � 0 , � , f s � 0 } p j , f i ∈ Q [ U 1 , � , U d , X d +1 , � , X n ] Results - parametric systems { p 1 = 0 , First geometrical analysis. Intrinsic object : the minimal discriminant variety wrt projection onto the parameters’s space (Lazard/Rouillier ad-hoc variants in 2002/2003 - formal definition in 2006) : An algebraic variety defining a partition of the parameter’s space : the variety itself and the cells defining its complementary in the projection of the studied constructible set. Over each cell, the solutions of the system are “easy to study” (constant number of solutions, analytic cover, etc.). Geometrical objects : projections of some subsets (singular points, crit- ical points of the projection, points from infinity, etc.), analytic covers. Computational tools : decompositions, elimination, saturation, homoge- nization (see algebraic solving). Next stage : compute 1 point per cell - eventually describe the cells - solve the related zero-dimensional systems (for each test point).
� , p r = 0 , f 1 � 0 , � , f s � 0 } p j , f i ∈ Q [ U 1 , � , U d , X d +1 , � , X n ] Results - general systems (n=0) { p 1 = 0 , One point per semi-algebraically connected component of a semi- algebraic set. Basic strategy : critical points of a Morse function Basic algorithms (real algebraic sets) : distance function (Aubry/Rouillier/Safey - 2002), coordinate function (Safey/Schost - 2003) Main improvements : Complementary of an hypersurface / Generalized critical values (Safey 2006) Singular hypersurfaces / Lagrange multipliers (Safey 2005) Computational tools : decompositions, elimination, saturation, homoge- nization (see algebraic solving).
� , p r = 0 , f 1 � 0 , � , f s � 0 } p j , f i ∈ Q [ U 1 , � , U d , X d +1 , � , X n ] Results - Zero-dimensional solving ( d = 0) � , X n = h n ( T ) � , f l = h l { p 1 = 0 , � , p r � ) and V ( f ) : preserves multiplicities and Basic object to be computed : Rational Univariate Representation � , X n ] � � ′ ( T ) ′ ( T ) f ( T ) = 0 , X 1 = h 1 ( T ) h ( T ) ,f 1 = h 1 � , p r � h ( T ) , g ( T ) , g ( T ) Bijection between V ( � p 1 , real roots. Q [ X 1 , Prerequisites : a description of (Gröbner bases, Triangular � p 1 , sets, generalized normal forms, etc.) Basic algorithms : Shape position ideals (Faugère - 1996) - General case (Rouillier 1999). Main improvements : extension for inequalities (Rouillier 2005) - direct computation for shape position ideals (Faugère - RR form) - Use of LLL (Ars/Faugère - 2004) - Imbricated systems in shape position (Faugère 2005) - Generalized normal forms (Trébuchet/Mourrain 2005 . BPA ISSAC 2005). Main computational tools : linear algebra
� , p r = 0 , f 1 � 0 , � , f s � 0 } p j , f i ∈ Q [ U 1 , � , U d , X d +1 , � , X n ] Results - univariate solving (d=0,n=1) { p 1 = 0 , Basic strategy : isolate the real roots by means of intervals with rational bounds. Basic algorithm : memory optimal variant of Uspensky’s method (Rouil- lier/Zimmermann 2003) Main improvements : use of multi-precision interval arithmetic (Rouil- lier/Zimmermann 2003 - Rouillier/Revol 2005) - particular case of a Rational Univariate representation (Rouillier 2004).
Main complexity results Gröbner bases : overconstraint systems (Bardet/Faugère/Salvy 2005) - zero-dimensional systems (Lazard/Hashemi 2006). Zero-dimensional systems : RUR (Rouillier 2005) Parametric systems : in the case of “well behaved” systems (Moroz 2006 - BPA ISSAC 2006) Positive dimensional systems : Bihomogeneous structure of Lagrange systems (Safey/Trébuchet 2006) - Generalized critical values (Safey 2006) - Singular hypersurfaces (Safey 2005).
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