Real Solving on Algebraic Systems of Small Dimension Masters Thesis - PowerPoint PPT Presentation

Real Solving on Algebraic Systems of Small Dimension Master’s Thesis Presentation Dimitrios I. Diochnos University of Athens March 8, 2007 D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 1 / 66

Outline Introduction 1 Results in Univariate Polynomials 2 Results in Bivariate Polynomials 3 Real Solving on Bivariate Systems 4 5 Applications 6 Implementation D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 2 / 66

Outline Introduction 1 Results in Univariate Polynomials 2 Results in Bivariate Polynomials 3 Real Solving on Bivariate Systems 4 5 Applications 6 Implementation D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 3 / 66

Notation and Conventions Complexity: � O B implies that we are ignoring (poly-)logarithmic factors. ◮ Length function: L () ◮ Given ν ∈ Z , L ( ν ) implies the bitsize of integer ν. ◮ Given A ∈ Z [ x ] L ( A ) implies the maximum bitsize of the coefficients. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 4 / 66

Notation and Conventions Complexity: � O B implies that we are ignoring (poly-)logarithmic factors. ◮ Length function: L () ◮ Given ν ∈ Z , L ( ν ) implies the bitsize of integer ν. ◮ Given A ∈ Z [ x ] L ( A ) implies the maximum bitsize of the coefficients. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 4 / 66

Operations on Lists. Sign Variations: Given a list of signs compute sign-swaps. Ignore zeros. Example VAR ([+ , + , − , 0 , − , 0 , 0 , +]) = 2 Intermediate Points: Given a list of (sorted) rational numbers compute rationals in between. Compute two more bounding rationals for the entire sequence. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 5 / 66

Operations on Lists. Sign Variations: Given a list of signs compute sign-swaps. Ignore zeros. Example VAR ([+ , + , − , 0 , − , 0 , 0 , +]) = 2 Intermediate Points: Given a list of (sorted) rational numbers compute rationals in between. Compute two more bounding rationals for the entire sequence. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 5 / 66

Polynomial GCD Computation Euclid’s Algorithm. ◮ Works fine when F , G ∈ Q [ x ] . What if we want to work in Z [ x ] ? ◮ Pseudo -divisions are required. k · F = Q · G + λ · R where F , G , Q , R ∈ Z [ x ] and k , λ ∈ Z . Pseudo-Euclidean:  ( lead ( G ) δ + 1 , 1 ) ( k , λ ) =  R = rem ( F , G )  Q = quo ( F , G ) D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 6 / 66

Polynomial GCD Computation Euclid’s Algorithm. ◮ Works fine when F , G ∈ Q [ x ] . What if we want to work in Z [ x ] ? ◮ Pseudo -divisions are required. k · F = Q · G + λ · R where F , G , Q , R ∈ Z [ x ] and k , λ ∈ Z . Pseudo-Euclidean:  ( lead ( G ) δ + 1 , 1 ) ( k , λ ) =  R = rem ( F , G )  Q = quo ( F , G ) D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 6 / 66

Polynomial GCD Computation Euclid’s Algorithm. ◮ Works fine when F , G ∈ Q [ x ] . What if we want to work in Z [ x ] ? ◮ Pseudo -divisions are required. k · F = Q · G + λ · R where F , G , Q , R ∈ Z [ x ] and k , λ ∈ Z . Pseudo-Euclidean:  ( lead ( G ) δ + 1 , 1 ) ( k , λ ) =  R = rem ( F , G )  Q = quo ( F , G ) D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 6 / 66

Sturm Sequences and Signed PRSs. Corollary Every sequence ( A i ) = ( A , A ′ , . . . ) where λ i A i = k i A i − 2 + A i − 1 Q i − 1 where k i , λ i ∈ R , k i λ i < 0 and A 1 = A square-free is a Sturm sequence in [ I l , I R ] where A ( I L ) A ( I R ) � = 0 . D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 7 / 66

Definitions on Signed PRSs. λ i A i = k i A i − 2 + A i − 1 Q i − 1 √ 2 ) i ) ⇒ Impractical. Pseudo-Euclidean: L ( A i ) = O (( 1 + Primitive-Part: 8 ( lead ( A i − 1 ) δ i + 1 , content ( prem ( A i , A i − 1 ))) < ( k i , λ i ) = A i = rem ( λ i A i − 2 , A i − 1 ) / k i : Q i = quo ( λ i A i − 2 , A i − 1 ) Time Bound: e O B ( p 2 q 2 τ ) . Output Bound: L ( A i ) = O ( p τ ) . SubResultant: ˘ A i = prem ( A i − 2 , A i − 1 ) / lead ( A i − 2 ) Time Bound: e O B ( p 2 q τ ) . Output Bound: L ( A i ) = O ( p τ ) . Sturm-Habicht: Similar to SubResultant sequences. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 8 / 66

Example on PRSs. Given: f = x 8 + x 6 − 3 x 4 − 3 x 3 + 8 x 2 + 2 x − 5 g = 3 x 6 + 5 x 4 − 4 x 2 − 9 x + 21 Euclidean: 15 x 4 − 3 x 2 + 9 15795 x 2 + 30375 x − 59535 1254542875143750 x − 1654608338437500 − 12593338795500743100931141992187500 Primitive Part: 5 x 4 − x 2 + 3 13 x 2 + 25 x − 49 4663 x − 6150 − 1 SubResultant: 15 x 4 − 3 x 2 + 9 65 x 2 + 125 x − 245 9326 x − 12300 − 260708 D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 9 / 66

Outline Introduction 1 Results in Univariate Polynomials 2 Bounds Real Algebraic Numbers Solving in One Variable Results in Bivariate Polynomials 3 Real Solving on Bivariate Systems 4 Applications 5 6 Implementation D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 10 / 66

Bounding Roots Let ̺ ∈ C be a root of the polynomial A ( x ) = a d x d + · · · + a 1 x + a 0 . Cauchy, Mignotte: 1 + max {| a 0 | , | a 1 | , . . . , | a d − 1 |} | ̺ | ≤ | a d | � � � � � d | a d − 1 | d | a d − 2 | d | a d − 3 | d | a 0 | 3 d | ̺ | ≤ max , , , . . . , | a d | | a d | | a d | | a d | Zassenhaus: �� � � � � � a k � � d − k | ̺ | ≤ 2 · max � � a d k Complexity: � O B ( d τ ) . D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 11 / 66

Real Algebraic Numbers Definition: Let A ∈ Z [ x ] . Each ̺ ∈ R such that A ( ̺ ) = 0 is a Real Algebraic Number and we will write ̺ ∈ R alg . Representation: Isolating Intervals. ̺ ∼ = [ square-free ( A ) , [ I L , I R ] ] such that I L , I R ∈ Q , I L ≤ ̺ ≤ I R and ̺ unique root in interval [ I L , I R ] . Basic Operations: Sign-At. Comparison. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 12 / 66

Real Algebraic Numbers Definition: Let A ∈ Z [ x ] . Each ̺ ∈ R such that A ( ̺ ) = 0 is a Real Algebraic Number and we will write ̺ ∈ R alg . Representation: Isolating Intervals. ̺ ∼ = [ square-free ( A ) , [ I L , I R ] ] such that I L , I R ∈ Q , I L ≤ ̺ ≤ I R and ̺ unique root in interval [ I L , I R ] . Basic Operations: Sign-At. Comparison. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 12 / 66

Real Algebraic Numbers Definition: Let A ∈ Z [ x ] . Each ̺ ∈ R such that A ( ̺ ) = 0 is a Real Algebraic Number and we will write ̺ ∈ R alg . Representation: Isolating Intervals. ̺ ∼ = [ square-free ( A ) , [ I L , I R ] ] such that I L , I R ∈ Q , I L ≤ ̺ ≤ I R and ̺ unique root in interval [ I L , I R ] . Basic Operations: Sign-At. Comparison. D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 12 / 66

Polynomial Evaluation at a R alg point. Input: α = [ A , [ L , R ]] and f ∈ R [ x ] ; A , f square-free Output: sign ( f ( α )) Compute SubResultant sequence. 1 Evaluate on endpoints. 2 Yield result with Sign-Variations. 3 Complexity: � O B ( pq τ + p min { p , q } 2 τ ) . D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 13 / 66

Comparison between R alg numbers. Input: α = [ A , [ L , R ]] and β = [ B , [ L , R ]] Output: Decide α ≤ β. Idea: Compute S IGN -A T ( A ( β )) . Note: We know the sign of A ′ ( α ) . Complexity: � O B ( pq τ + p min { p , q } 2 τ ) . D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 14 / 66

Solving in One Variable Input: A ∈ Z [ x ] , square-free. Output: A list S of intervals that contain the real roots of A . Subdivision Method. Complexity: [Emiris,Mourrain,Tsigaridas - 2006] O B ( p 6 + p 4 τ 2 ) (+ multiplicities) Time: � Output: O ( p τ ) D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 15 / 66

Outline Introduction 1 Results in Univariate Polynomials 2 Results in Bivariate Polynomials 3 Extensions. Resultant Real Solving on Bivariate Systems 4 Applications 5 Implementation 6 D. I. Diochnos (Univ. of Athens, µ Q λ ∀ ) Real Solving on Bivariate Algebraic Systems Mar ’07 16 / 66