Gröbner Cover Canonical discussion of polynomial systems with parameters Antonio Montes Universitat Politècnica de Catalunya Logroño, 9-11-2010 Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 1 / 51
References Based on Antonio Montes, Michael Wibmer. “Gröbner Bases for Polynomial Systems with Parameters". Journal of Symbolic Computation 45 (2010) 1391 - 1425. Antonio Montes, Tomás Recio. “Generalization of Steiner-Lehmus Theorem using the Gröbner Cover". Work in progress. Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 2 / 51
Outline Parametric polynomial discussion 1 Existence of the Gröbner cover 2 The Gröbner Cover algorithm 3 Applications 4 Automatic Discovery of Geometric Theorems Generalizing the Steiner-Lehmus Theorem Casas conjecture Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 3 / 51
Index Parametric polynomial discussion 1 Existence of the Gröbner cover 2 The Gröbner Cover algorithm 3 Applications 4 Automatic Discovery of Geometric Theorems Generalizing the Steiner-Lehmus Theorem Casas conjecture Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 4 / 51
Parametric polynomial discussion Goal Data: Parametric polynomial system of equations p 1 ( a 1 , . . . , a m , x 1 , . . . , x n ) = 0 · · · p r ( a 1 , . . . , a m , x 1 , . . . , x n ) = 0 Goal: describe the different kind of solutions ( x 1 , . . . , x n ) in dependence of the parameters a 1 , · · · , a m . Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 5 / 51
Some notations Let: K be a computable field (in practice Q ). K be an algebraically closed extension of K (in practice C ). K [ a ] the polynomial ring in the parameters a = a 1 , . . . , a m over K . K [ a ][ x ] the polynomial ring in the variables x = x 1 , . . . , x n over K [ a ] . K m is the parameter space. Fix: ≻ x monomial ordering wrt x and the ideal I = � p 1 ( a , x ) , · · · , p r ( a , x ) � ⊂ K [ a ][ x ] lpp ( G ) = set of leading power products wrt ≻ x of the polynomials in G . Specialization: m ) ∈ K m a = ( a 0 1 , · · · , a 0 I a = � p 1 ( a , x ) , · · · , p r ( a , x ) � ⊂ K [ x ] Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 6 / 51
Goal Gröbner bases are the computational method par excellence for studying polynomial systems. The set of lpp of the reduced Gröbner basis determines the type of solutions of the system. In the case of parametric polynomial systems the goal is to describe the reduced Gröbner basis of I a ⊂ K [ x ] (with respect to ≻ x ) in dependence of a ∈ K m . Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 7 / 51
Precedents Weispfenning (1992) Given I = � p 1 , . . . , p r � ⊂ K [ a ][ x ] = K [ a , x ] and ≻ x A Comprehensive Gröbner System (CGS) for I and ≻ x is a finite set of pairs { ( S 1 , B 1 ) , . . . , ( S s , B s ) } (Segments: S i , Bases: B i ) such that The S i ’s are constructible subsets of K m such that K m = ∪ S i . 1 The B i ’s are finite subsets of K ( a )[ x ] and B i ( a ) = { p ( a , x ) : p ∈ B i } 2 is a Gröbner basis of I a with respect to ≻ x for every a ∈ S i . B i ⊂ I . Leads to a Comprehensive Gröbner Basis Faithful: B i reduced. Non-faithful: Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 8 / 51
Historical development Two directions: Speed up. Duval (1995), Dellière (1999), Kapur (1995), Kalkbrenner (1997), Sato (2003), Suzuki & Sato (2006), Nabeshima (2006), Deepak Kapur & Yao Sun & Dingkang Wang (2010). Improve output. Montes (2002), Weispfenning (2003), Wibmer (2007), Manubens & Montes (2009), Montes & Wibmer (2010). Our goal: best output for applications, disjoint segments, segments with constant lpp , minimal number of segments, canonical output, if possible, locally closed segments. Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 9 / 51
A simple but critical example Consider the ideal F = � ax + by , cx + dy � . It is elementary to obtain the following discussion ( lex ( x , y ) ): lpp Num. segment basis C 4 \ V ( ad − bc ) [ y , x ] [ y , x ] 1 � b a , d V ( ad − bc ) \ V ( a , c ) [ x + y ] [ x ] � 2 c V ( a , c ) \ V ( a , b , c , d ) [ y ] [ y ] 3 V ( a , b , c , d ) 4 [ ] [ ] To summarize into a unique segment each set of solotions with the same lpp , ordinary polynomials are not sufficient. We need more general functions: f : S = V ( ad − bc ) \ V ( a , b ) − → regular functions C � b a , d ( a , b , c , d ) � �→ c I -regular functions g : S = V ( ad − bc ) \ V ( a , b ) C [ x , y ] − → ( a , b , c , d ) x + fy �→ Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 10 / 51
Index Parametric polynomial discussion 1 Existence of the Gröbner cover 2 The Gröbner Cover algorithm 3 Applications 4 Automatic Discovery of Geometric Theorems Generalizing the Steiner-Lehmus Theorem Casas conjecture Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 11 / 51
Locally closed sets Definition A subset S ⊂ K m is locally closed, if it is difference of two varieties: S = V ( M ) \ V ( N ) . Definition (Open subset) A subset U ⊂ S is said to be open on S if S \ U � S . Proposition (Canonical representation) Let S ⊂ K m be a locally closed set. Then, there exist uniquely determined radical ideals a ⊂ b of K [ a ] , with S = V ( a ) \ V ( b ) , such that S = V ( a ) , S \ S = V ( b ) . The pair ( a , b ) -top,hole- is called the canonical representation of S . Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 12 / 51
Locally closed sets Proposition (Canonical prime representation) Let S ⊂ K m be a locally closed set. Then, there exist a unique canonical prime representation of S given the prime components of a , say p i , and associated to each, a set of prime ideals p ij (holes) in the form (( p 1 , ( p 11 , . . . , p 1 j 1 )) , . . . , ( p k , ( p k 1 , . . . , p kj k ))) so that j i k � � . S = V ( p i ) \ ( V ( p ij )) i = 1 j = 1 and p i ⊂ p ij for all i , j , such that S = V ( p 1 ) ∪ . . . ∪ V ( p r ) and ( S \ S ) ∩ V ( p i ) = V ( p i 1 ) ∪ · · · ∪ V ( p ir i ) are the minimal decompositions into irreducible closed sets. Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 13 / 51
I -regular functions Definition ( I -Regular function) Let S be a locally closed subset of K m . We call a function f : S − → K [ x ] I -regular, if ∀ a ∈ S it exists an open U ⊂ S with a ∈ U and f ( b ) = P ( b , x ) for all b ∈ U , Q ( b ) where P ∈ I and Q ∈ K [ a ] and Q ( b ) � = 0 for all b ∈ U . Remark Let P and Q be a polynomials as above, (they are not unique), S = V ( a ) \ V ( b ) and p ( b , x ) = P ( b , x ) mod a . If f is monic and lpp ( f ) is constant on S , then, for all b ∈ U is lpp x ( p ( b , x )) = lpp x ( f ) , and lc x ( p ( b , x )) = Q ( b ) mod a . Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 14 / 51
Parametric subsets m ) Definition (Parametric subset of K A locally closed subset S ∈ K m is called parametric (wrt to I and ≻ x ) if there exist monic I -regular functions { g 1 , . . . , g s } over S so that { g 1 ( a , x ) , . . . , g s ( a , x ) } is the reduced Gröbner basis of I a for all a ∈ S . Note Note that the definition immediately implies that if a , b lie in a parametric set S , then lpp x ( I a ) = lpp x ( I b ) . The amazing thing is that the converse also holds if we additionally assume that I ⊂ K [ a ][ x ] is homogeneous (wrt to the variables). Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 15 / 51
Wibmer’s Theorem Theorem (M. Wibmer) Let I ⊂ K [ a ][ x ] be a homogeneous ideal and a ∈ K m . Then the set S a = { b ∈ K m : lpp x ( I b ) = lpp x ( I a ) } is parametric. In particular, S a is locally closed. Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 16 / 51
Gröbner cover Definition (Gröbner cover) By a Gröbner cover of K m wrt to I and ≻ x we mean a finite set of pairs { ( S 1 , B 1 ) , . . . , ( S r , B r ) } such that the S i ’s are parametric and so, B i ⊂ O ( S i )[ x ] is the reduced 1 Gröbner basis of I over S i for i = 1 , . . . , r , and the union of all S i ’s equals K m . 2 Theorem (Canonical Gröbner cover) Let I ⊂ K [ a ][ x ] be a homogeneous ideal. Then there exists a unique Gröbner cover of K m with minimal cardinality which we call the canonical Gröbner cover. It is disjoint and two points a , b ∈ K m lie in the same segment if and only if lpp x ( I a ) = lpp x ( I b ) . Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 17 / 51
Gröbner cover Definition (Gröbner cover) By a Gröbner cover of K m wrt to I and ≻ x we mean a finite set of pairs { ( S 1 , B 1 ) , . . . , ( S r , B r ) } such that the S i ’s are parametric and so, B i ⊂ O ( S i )[ x ] is the reduced 1 Gröbner basis of I over S i for i = 1 , . . . , r , and the union of all S i ’s equals K m . 2 Theorem (Canonical Gröbner cover) Let I ⊂ K [ a ][ x ] be a homogeneous ideal. Then there exists a unique Gröbner cover of K m with minimal cardinality which we call the canonical Gröbner cover. It is disjoint and two points a , b ∈ K m lie in the same segment if and only if lpp x ( I a ) = lpp x ( I b ) . Antonio Montes (UPC) Gröbner Cover Logroño-2010, 9-11-2010 17 / 51
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