An Efficient Algorithm for Computing Parametric Multivariate - PowerPoint PPT Presentation

An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD Dong Lu Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science, CAS Joint work with Deepak Kapur, Michael Monagan, Yao Sun and Dingkang Wang July 16-19, 2018, The City University of New York, USA Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 1 / 28

Outline 1 The Problem 2 Previous Works 3 The New Algorithm 4 A Simple Example 5 Implementation 6 Conclusions Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 2 / 28

Outline 1 The Problem 2 Previous Works 3 The New Algorithm 4 A Simple Example 5 Implementation 6 Conclusions Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 3 / 28

The Problem Notations k is a field. e 1 := (1 , 0) and e 2 := (0 , 1). X = { x 1 , . . . , x n } are variables. U = { u 1 , . . . , u m } are parameters. k [ U ][ X ] is the polynomial ring in X . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 4 / 28

The Problem Example 1 Let f 1 , f 2 ∈ C [ a ][ x , y ], where f 1 = ax 3 + ( a 3 − a + 1) x 2 y + ( a 2 + 2) xy 2 + (3 a 2 − 3) y 3 , � f 2 = ax 3 + ( a + 1) x 2 y + 4 xy 2 + 3 y 3 . The results are:  when a = 0 , gcd( f 1 , f 2 ) = y ( x + 3 y );   when a 2 − 2 = 0 , gcd( f 1 , f 2 ) = 2 x 3 + ( a + 2) x 2 y + 4 axy 2 + 3 ay 3 ; when a ( a 2 − 2) � = 0 , gcd( f 1 , f 2 ) = ax 2 + xy + 3 y 2 .   Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 5 / 28

The Problem Problem: For any given parametric polynomials f 1 , f 2 , . . . , f s in k [ U ][ X ], how to divide the parametric space and obtain the corresponding GCD on each branch quickly? Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 6 / 28

Outline 1 The Problem 2 Previous Works 3 The New Algorithm 4 A Simple Example 5 Implementation 6 Conclusions Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 7 / 28

Previous Works • Non-parametric polynomials case J. Moses and D.Y.Y. Yun: The EZ GCD algorithm. In Proceedings of ACM’ 73 , ACM Press, New York, 1973, 159-166. (Hensel lifting) R. Zippel: Probabilistic algorithms for sparse polynomials. In Proceedings of EUROSAM’ 79 , Springer-Verlag, 1979, 216-226. (sparse interpolation) P. Gianni and B. Trager: GCDs and factoring multivariate polynomials using Gr¨ obner bases. In Proceedings of EUROCAL’ 85 , Springer, Berlin, Heidelberg, 1985, 409-410. (Gr¨ obner basis) T. Sasaki and M. Suzuki: Three new algorithms for multivariate polynomial GCD. Journal of Symbolic Computation , 1992, 395-411. (Gr¨ obner basis) Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 8 / 28

Previous Works • Parametric polynomials case S.A. Abramov and K.Y. Kvashenko: On the greatest common divisor of polynomials which depend on a parameter. In Proceedings of ISSAC 1993 , 152-156. (subresultant chain) A. Ayad: Complexity of algorithms for computing greatest common divisors of parametric univariate polynomials. International Journal of Algebra , 2010, 173-188. (Gaussian elimination) K. Nagasaka: Parametric greatest common divisors using comprehensive Gr¨ obner systems. In Proceedings of ISSAC 2017 , 341-348. (extended the ideas of Gianni and Trager as well as Sasaki and Suzuki) Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 9 / 28

Outline 1 The Problem 2 Previous Works 3 The New Algorithm 4 A Simple Example 5 Implementation 6 Conclusions Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 10 / 28

The New Algorithm • Non-parametric polynomials case Definition 1 Let I ⊂ k [ X ] be an ideal and g ∈ k [ X ] \ { 0 } . The set I : g = { f ∈ k [ X ] : fg ∈ I } is called the quotient ideal (or colon ideal ) of I divided by g . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 11 / 28

The New Algorithm • Non-parametric polynomials case Definition 1 Let I ⊂ k [ X ] be an ideal and g ∈ k [ X ] \ { 0 } . The set I : g = { f ∈ k [ X ] : fg ∈ I } is called the quotient ideal (or colon ideal ) of I divided by g . Example 2 In k [ x , y , z ], let I = � xy � and g = xz . Then I : g = { f : xzf ∈ � xy �} = { f : zf ∈ � y �} = � y � . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 11 / 28

The New Algorithm Lemma 1 Let f 1 , f 2 ∈ k [ X ] \ { 0 } , then � f 1 � : f 2 is a principal ideal. Suppose that � f 1 � : f 2 = � f � , then we have gcd( f 1 , f 2 ) = f 1 f . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 12 / 28

The New Algorithm Lemma 1 Let f 1 , f 2 ∈ k [ X ] \ { 0 } , then � f 1 � : f 2 is a principal ideal. Suppose that � f 1 � : f 2 = � f � , then we have gcd( f 1 , f 2 ) = f 1 f . Example 3 Let f 1 = xy and f 2 = xz . According to the Example 2, we have that � xy � : xz = � y � . Then gcd( f 1 , f 2 ) = f 1 / f = x . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 12 / 28

The New Algorithm Problem : How to compute the generator of � f 1 � : f 2 ? Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 13 / 28

The New Algorithm Problem : How to compute the generator of � f 1 � : f 2 ? ◮ Original Method Lemma 2 Let f 1 , f 2 ∈ k [ X ] \ { 0 } and w be a new variable, then � wf 1 , ( w − 1) f 2 � ∩ k [ X ] = � g � for some g ∈ k [ X ] \ { 0 } . Moreover, we have � f 1 � : f 2 = � g f 2 � and gcd( f 1 , f 2 ) = f 1 f 2 g . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 13 / 28

The New Algorithm ◮ New Method Lemma 3 Let f 1 , f 2 ∈ k [ X ] \ { 0 } and ≺ be a monomial order on k [ X ] 2 with e 2 ≺ e 1 . Suppose G is a minimal Gr¨ obner basis of � f 1 · e 1 , f 2 · e 1 − e 2 � . Then there is a unique f ∈ k [ X ] \ { 0 } such that f · e 2 ∈ G and � f 1 � : f 2 = � f � . Therefore, gcd( f 1 , f 2 ) = f 1 f . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 14 / 28

The New Algorithm ◮ New Method Lemma 3 Let f 1 , f 2 ∈ k [ X ] \ { 0 } and ≺ be a monomial order on k [ X ] 2 with e 2 ≺ e 1 . Suppose G is a minimal Gr¨ obner basis of � f 1 · e 1 , f 2 · e 1 − e 2 � . Then there is a unique f ∈ k [ X ] \ { 0 } such that f · e 2 ∈ G and � f 1 � : f 2 = � f � . Therefore, gcd( f 1 , f 2 ) = f 1 f . Example 4 Let f 1 = xy , f 2 = xz . Given the lexicographic order ≺ and extend it to k [ X ] 2 in a position over term with e 2 ≺ e 1 . A minimal Gr¨ obner basis of � xy · e 1 , xz · e 1 − e 2 � is G = { y · e 2 , xy · e 1 , xz · e 1 − e 2 } . So, f = y and � xy � : xz = � y � . Moreover, gcd( xy , xz ) = f 1 / f = x . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 14 / 28

The New Algorithm • Parametric polynomials case Theorem 1 Let f 1 , f 2 ∈ k [ U ][ X ] and ≺ be a monomial order on k [ X ] 2 with e 2 ≺ e 1 . Suppose { ( A i , G i ) } l i =1 is a minimal comprehensive Gr¨ obner system of � f 1 · e 1 , f 2 · e 1 − e 2 � . For each branch ( A i , G i ) , let H i = { f | f · e 2 ∈ G i } . Then we have 1 If H i = ∅ , then f 1 = 0 and gcd( f 1 , f 2 ) = f 2 on A i . 2 If H i � = ∅ , then H i = { f } and gcd( f 1 , f 2 ) = f 1 f on A i . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 15 / 28

The New Algorithm ◮ Parametric GCD Algorithm Input: f 1 , f 2 ∈ k [ U ][ X ], a constructible set A ⊂ ¯ k m , and two monomial orders ≺ X , ≺ U . Output: a comprehensive GCDs: { ( A i , h i ) } s i =1 , where h i = gcd ( f 1 , f 2 ) under any specialization from A i and ∪ s i =1 A i = A . obner system { ( A i , G i ) } s Step 1 : compute a minimal comprehensive Gr¨ i =1 for the module � f 1 · e 1 , f 2 · e 1 − e 2 � on A . Step 2 : For every i , let H i = { h | h · e 2 ∈ G i } , then do Step 2.1 : if H i is empty, then h i = f 2 on A i and turn to Step 2; otherwise, turn to Step 2.2. Step 2.2 : h i = f 1 / h on A i . Step 3 : return { ( A i , h i ) } s i =1 . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 16 / 28

Outline 1 The Problem 2 Previous Works 3 The New Algorithm 4 A Simple Example 5 Implementation 6 Conclusions Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 17 / 28

A Simple Example Example 5 Let f 1 , f 2 , f 3 ∈ C [ a , b ][ x , y , z ] be as follows: f 1 = ax 2 + bxy + a 2 xz + abx + abyz + b 2 y ,   f 2 = ax 2 + bxy + ( ab − a ) xz − a 2 x + ( b 2 − b ) yz − aby , f 3 = ax 2 + bxy + a 2 xz + ( a 2 − ab ) x + abyz + ( ab − b 2 ) y ,  Using the new algorithm to compute the GCDs of f 1 , f 2 , f 3 . Dong Lu (KLMM, AMSS, CAS) Algorithm for Computing Parametric GCD July 16-19, CUNY, New York 18 / 28