Irredundant Triangular Decomposition Gleb Pogudin 1 , Agnes Szanto 2 - PowerPoint PPT Presentation

Irredundant Triangular Decomposition Gleb Pogudin 1 , Agnes Szanto 2 1 New York University and City University of New York 2 North Carolina State University

Big picture Question How can one represent the set W = { z ∈ C n | f 1 ( z ) = . . . = f m ( z ) = 0 } , where f 1 , . . . , f m ∈ C [ z 1 , . . . , z n ]? 1

Big picture Question How can one represent the set W = { z ∈ C n | f 1 ( z ) = . . . = f m ( z ) = 0 } , where f 1 , . . . , f m ∈ C [ z 1 , . . . , z n ]? Possible approaches • Gr¨ obner bases • Geometric resolution • Triangular decomposition • Witness sets 1

Big picture Question How can one represent the set W = { z ∈ C n | f 1 ( z ) = . . . = f m ( z ) = 0 } , where f 1 , . . . , f m ∈ C [ z 1 , . . . , z n ]? Possible approaches • Gr¨ obner bases • Geometric resolution • Triangular decomposition ← this talk • Witness sets 1

What is a triangular set? 2

What is a triangular set? Example (row echelon form) p 1 = x 1 − 2 x 2 + x 3 ∈ C [ x 1 , x 2 , x 3 ] p 2 = 5 x 1 − x 2 ∈ C [ x 1 , x 2 ] p 3 = x 1 − 1 ∈ C [ x 1 ] 2

What is a triangular set? Example (row echelon form) p 1 = x 1 − 2 x 2 + x 3 ∈ C [ x 1 , x 2 , x 3 ] p 2 = 5 x 1 − x 2 ∈ C [ x 1 , x 2 ] p 3 = x 1 − 1 ∈ C [ x 1 ] Example (nonlinear case) p 1 = x 1 x 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] 2 p 2 = x 3 2 − x 2 ∈ C [ x 1 , x 2 ] 1 x 2 and x 3 are leading variables and x 1 is a free variable . 2

What is a triangular set? Example (row echelon form) p 1 = x 1 − 2 x 2 + x 3 ∈ C [ x 1 , x 2 , x 3 ] p 2 = 5 x 1 − x 2 ∈ C [ x 1 , x 2 ] p 3 = x 1 − 1 ∈ C [ x 1 ] Example (nonlinear case) p 1 = x 1 x 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] 2 p 2 = x 3 2 − x 2 ∈ C [ x 1 , x 2 ] 1 x 2 and x 3 are leading variables and x 1 is a free variable . Remark extra assumption regular chain = triangular set + (“leading coefficient � = 0”) 2

How do we use regular chains? Pseudo-reduction! Input • regular chain { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ⊂ C [ x 1 , x 2 , x 3 ] • polynomial x 2 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] 3

How do we use regular chains? Pseudo-reduction! Input • regular chain { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ⊂ C [ x 1 , x 2 , x 3 ] vanish on the curve t → ( t 3 , t 2 , t ) ; • polynomial x 2 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] also vanishes on t → ( t 3 , t 2 , t ) . 3

How do we use regular chains? Pseudo-reduction! Input • regular chain { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ⊂ C [ x 1 , x 2 , x 3 ] vanish on the curve t → ( t 3 , t 2 , t ) ; • polynomial x 2 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] also vanishes on t → ( t 3 , t 2 , t ) . Pseudo-reduction 3

How do we use regular chains? Pseudo-reduction! Input • regular chain { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ⊂ C [ x 1 , x 2 , x 3 ] vanish on the curve t → ( t 3 , t 2 , t ) ; • polynomial x 2 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] also vanishes on t → ( t 3 , t 2 , t ) . Pseudo-reduction x 1 ( x 2 3 − x 2 ) − x 3 ( x 1 x 3 − x 2 r 1 := − x 1 x 2 + x 2 2 ) = r 1 → 2 x 3 , 3

How do we use regular chains? Pseudo-reduction! Input • regular chain { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ⊂ C [ x 1 , x 2 , x 3 ] vanish on the curve t → ( t 3 , t 2 , t ) ; • polynomial x 2 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] also vanishes on t → ( t 3 , t 2 , t ) . Pseudo-reduction x 1 ( x 2 3 − x 2 ) − x 3 ( x 1 x 3 − x 2 r 1 := − x 1 x 2 + x 2 2 ) = r 1 → 2 x 3 , x 1 ( x 2 2 x 3 − x 1 x 2 ) − x 2 2 ( x 1 x 3 − x 2 r 2 := − x 2 1 x 2 + x 4 2 ) = r 2 → 2 , 3

How do we use regular chains? Pseudo-reduction! Input • regular chain { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ⊂ C [ x 1 , x 2 , x 3 ] vanish on the curve t → ( t 3 , t 2 , t ) ; • polynomial x 2 3 − x 2 ∈ C [ x 1 , x 2 , x 3 ] also vanishes on t → ( t 3 , t 2 , t ) . Pseudo-reduction x 1 ( x 2 3 − x 2 ) − x 3 ( x 1 x 3 − x 2 r 1 := − x 1 x 2 + x 2 2 ) = r 1 → 2 x 3 , x 1 ( x 2 2 x 3 − x 1 x 2 ) − x 2 2 ( x 1 x 3 − x 2 r 2 := − x 2 1 x 2 + x 4 2 ) = r 2 → 2 , ( x 4 2 − x 2 1 x 2 ) − x 2 ( x 3 2 − x 2 1 ) = r 3 → r 3 := 0 3

What is a triangular decomposition? The ideal and variety defined by a regular chain Let ∆ ⊂ C [ x ] be a regular chain, then I (∆) := { f ∈ C [ x ] | f pseudo-reduces to zero w.r.t. ∆ } . 4

What is a triangular decomposition? The ideal and variety defined by a regular chain Let ∆ ⊂ C [ x ] be a regular chain, then I (∆) := { f ∈ C [ x ] | f pseudo-reduces to zero w.r.t. ∆ } . For example, I ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) = ( x 1 x 3 − x 2 2 , x 2 x 3 − x 1 , x 2 3 − x 2 ). 4

What is a triangular decomposition? The ideal and variety defined by a regular chain Let ∆ ⊂ C [ x ] be a regular chain, then I (∆) := { f ∈ C [ x ] | f pseudo-reduces to zero w.r.t. ∆ } . For example, I ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) = ( x 1 x 3 − x 2 2 , x 2 x 3 − x 1 , x 2 3 − x 2 ). Then V (∆) ⊂ C n is the set of common zeros of I (∆). 4

What is a triangular decomposition? The ideal and variety defined by a regular chain Let ∆ ⊂ C [ x ] be a regular chain, then I (∆) := { f ∈ C [ x ] | f pseudo-reduces to zero w.r.t. ∆ } . For example, I ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) = ( x 1 x 3 − x 2 2 , x 2 x 3 − x 1 , x 2 3 − x 2 ). Then V (∆) ⊂ C n is the set of common zeros of I (∆). Caveat: In general, I (∆) is larger than the ideal generated by ∆. 4

What is a triangular decomposition? The ideal and variety defined by a regular chain Let ∆ ⊂ C [ x ] be a regular chain, then I (∆) := { f ∈ C [ x ] | f pseudo-reduces to zero w.r.t. ∆ } . For example, I ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) = ( x 1 x 3 − x 2 2 , x 2 x 3 − x 1 , x 2 3 − x 2 ). Then V (∆) ⊂ C n is the set of common zeros of I (∆). Triangular decomposition Let X ⊂ C n be an algebraic variety. Then a representation X = V (∆ 1 ) ∪ . . . ∪ V (∆ m ) , where ∆ 1 , . . . , ∆ m are regular chains, is called a triangular decomposition of X . 4

What is a triangular decomposition? The ideal and variety defined by a regular chain Let ∆ ⊂ C [ x ] be a regular chain, then I (∆) := { f ∈ C [ x ] | f pseudo-reduces to zero w.r.t. ∆ } . For example, I ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) = ( x 1 x 3 − x 2 2 , x 2 x 3 − x 1 , x 2 3 − x 2 ). Then V (∆) ⊂ C n is the set of common zeros of I (∆). Triangular decomposition Let X ⊂ C n be an algebraic variety. Then a representation X = V (∆ 1 ) ∪ . . . ∪ V (∆ m ) , where ∆ 1 , . . . , ∆ m are regular chains, is called a triangular decomposition of X . Important remark : m = 1 is not enough for an arbitrary variety. 4

Irredundant triangular decomposition Definition A triangular decomposition X = V (∆ 1 ) ∪ . . . ∪ V (∆ m ) is called irredundant if ∀ ( i � = j ) ∀ ( C = an irreducible component of V (∆ i )) C �⊂ V (∆ j ) . 5

Irredundant triangular decomposition Definition A triangular decomposition X = V (∆ 1 ) ∪ . . . ∪ V (∆ m ) is called irredundant if ∀ ( i � = j ) ∀ ( C = an irreducible component of V (∆ i )) C �⊂ V (∆ j ) . Main problem The main problem is to design a good algorithm such that Input An algebraic variety defined by a system of polynomial equations f 1 = . . . = f s = 0 Output Irredundant triangular decomposition of X 5

Motivation • Reduce the size of the output 6

Motivation • Reduce the size of the output • Get correct information about the geometry of a variety 6

Motivation • Reduce the size of the output • Get correct information about the geometry of a variety Example System of equations x 1 x 3 − x 2 2 = x 2 x 3 − x 1 = x 2 3 − x 2 = 0 . 6

Motivation • Reduce the size of the output • Get correct information about the geometry of a variety Example System of equations x 1 x 3 − x 2 2 = x 2 x 3 − x 1 = x 2 3 − x 2 = 0 . Irredundant decomposition X = V ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) 6

Motivation • Reduce the size of the output • Get correct information about the geometry of a variety Example System of equations x 1 x 3 − x 2 2 = x 2 x 3 − x 1 = x 2 3 − x 2 = 0 . Irredundant decomposition X = V ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) Decomposition by RegularChains ( Maple ) X = V ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) ∪ V ( { x 3 , x 2 , x 1 } ) 6

Motivation • Reduce the size of the output • Get correct information about the geometry of a variety Example System of equations x 1 x 3 − x 2 2 = x 2 x 3 − x 1 = x 2 3 − x 2 = 0 . Irredundant decomposition X = V ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) Decomposition by RegularChains ( Maple ) X = V ( { x 1 x 3 − x 2 2 , x 3 2 − x 2 1 } ) ∪ V ( { x 3 , x 2 , x 1 } ) • Design better Hensel lifting-based algorithms (later in the talk) 6

State of the art 7

State of the art Theoretical grounds and first algorithms due to Ritt, Wu, Lazard, Aubry, Kalkbrenner , and other researchers 7

State of the art Theoretical grounds and first algorithms due to Ritt, Wu, Lazard, Aubry, Kalkbrenner , and other researchers General algorithms without irredundancy guarantees • General theoretical algorithm ( 1999 ) Szanto • Maple package RegularChains ( 2005 ) Alvandi, Chen, Lemaire, Moreno Maza, Xie, ... 7