Real Solving on Algebraic Systems of Small Dimension

Transcription

1 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

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

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

4 Notation and Conventions Complexity: Õ 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

5 Notation and Conventions Complexity: Õ 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

6 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

7 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

8 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: (k, λ) = (lead (G) δ+1, 1) 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

9 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: (k, λ) = (lead (G) δ+1, 1) 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

10 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: (k, λ) = (lead (G) δ+1, 1) 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

11 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

12 Definitions on Signed PRSs. λ i A i = k i A i 2 + A i 1 Q i 1 Pseudo-Euclidean: L(A i ) = O((1 + 2) i ) Impractical. Primitive-Part: 8 < SubResultant: : (k i, λ i ) = (lead (A i 1 ) δ i +1, content (prem (A i, A i 1 ))) 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 OB (p 2 q 2 τ). Output Bound: L(A i ) = O(pτ). Ai = prem (A i 2, A i 1 ) / lead (A i 2 ) Time Bound: e OB (p 2 qτ). Output Bound: L(A i ) = O(pτ). 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

13 Example on PRSs. Given: f = x 8 + x 6 3x 4 3x 3 + 8x 2 + 2x 5 g = 3x 6 + 5x 4 4x 2 9x + 21 Euclidean: 15x 4 3x x x x Primitive Part: 5x 4 x x x x SubResultant: 15x 4 3x x x x D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar 07 9 / 66

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

15 Bounding Roots Let ϱ C be a root of the polynomial A(x) = a d x d + + a 1 x + a 0. Cauchy, Mignotte: ϱ 1 + max{ a 0, a 1,..., a d 1 } a d { d a ϱ max d 1 d a, d 2, 3 a d a d d a d 3,..., d a d } d a 0 a d Zassenhaus: ϱ 2 max k { d k a k a d } Complexity: Õ B (dτ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

16 Real Algebraic Numbers Definition: Let A Z[x]. Each ϱ R such that A(ϱ) = 0 is a and we will write ϱ R alg. Representation: Isolating Intervals. Real Algebraic Number ϱ = [ square-free(a), [ I L, I R ] ] such that I L, I R Q, I L ϱ I R and ϱ 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 / 66

17 Real Algebraic Numbers Definition: Let A Z[x]. Each ϱ R such that A(ϱ) = 0 is a and we will write ϱ R alg. Representation: Isolating Intervals. Real Algebraic Number ϱ = [ square-free(a), [ I L, I R ] ] such that I L, I R Q, I L ϱ I R and ϱ 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 / 66

18 Real Algebraic Numbers Definition: Let A Z[x]. Each ϱ R such that A(ϱ) = 0 is a and we will write ϱ R alg. Representation: Isolating Intervals. Real Algebraic Number ϱ = [ square-free(a), [ I L, I R ] ] such that I L, I R Q, I L ϱ I R and ϱ 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 / 66

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

20 Comparison between R alg numbers. Input: α = [A, [L, R]] and β = [B, [L, R]] Output: Decide α β. Idea: Compute SIGN-AT(A(β)). Note: We know the sign of A (α). Complexity: Õ B (pqτ + p min{p, q} 2 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

21 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 ] Time: Õ B (p 6 + p 4 τ 2 ) (+ multiplicities) Output: O(pτ) D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

23 Bivariate SignAt. Input: α = [A, [A L, A R ]], β = [B, [B L, B R ]] and f Z[x, y]. Output: Compute the sign of f (α, β). 1 Compute a Sturm sequence of (A, f ). 2 Evaluate the sequence on each of endpoints A L, A R of α. 3 Perform SIGN-AT at y = β on each polynomial in sequence. 4 Count sign-variations on sequences. 5 Yield result with (V L V R ) A (α) Complexity: Sign can be computed in ÕB(n1 2n3 2 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

24 Working in Extension-Fields. GCD Computation [Hoeij and Monagan ] Other required operations are reduced to univariate SIGN-AT. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

25 Resultant Theorem Given f, g Z[x] such that f = n i=1 a ix x and g = m j=1 b jx j with a n b m 0, there is a unique (up to sign) irreducible polynomial res(f, g) Z[a n,..., a 0, b m,..., b 0 ] which is zero iff f, g have a common factor. It is homogeneous and deg(res(f, g)) = deg(f ) + deg(g) = n + m. This polynomial is called resultant. Theorem Given f, g K[x] we can compute the resultant res(f, g) via the Sylvester matrix Syl(f, g). More specifically, we have: Complexity: Õ B (nmτ). res(f, g) = det(syl(f, g)). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

26 Resultant Definition (Sylvester Matrix) The Sylvester matrix of f and g is the (n + m) (n + m) matrix defined as follows: a n a n 1 a n a 0 a n a n 1 a n a a n a n 1 a n a 0 Syl(f, g) = b m b m 1 b m 2... b 0 b m b m 1 b m 2... b b m b m 1 b m 2... b 0 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

27 Resultant D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

29 The Problem Given { f (x, y) = d 1 i=1 a ix i g(x, y) = d 2 j=1 b jx j such that f, g Z[x, y] we want to compute all such that (X 0, Y 0 ) R alg f (X 0, Y 0 ) = g(x 0, Y 0 ) = 0. Assuming d f, d g N and L(f ), L(g) N previous complexity bounds were: Isolating Intervals: Õ B (N 30 ) [Arnon, McCallum ] Thom s Encoding: Õ B (N 16 ) [González-Vega, Kahoui ] D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

30 The Problem Given { f (x, y) = d 1 i=1 a ix i g(x, y) = d 2 j=1 b jx j such that f, g Z[x, y] we want to compute all such that (X 0, Y 0 ) R alg f (X 0, Y 0 ) = g(x 0, Y 0 ) = 0. Assuming d f, d g N and L(f ), L(g) N previous complexity bounds were: Isolating Intervals: Õ B (N 30 ) [Arnon, McCallum ] Thom s Encoding: Õ B (N 16 ) [González-Vega, Kahoui ] D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

31 Our Work [D, Emiris, Tsigaridas ] Running Time: Õ B (N 12 ) Output Size: Õ B (N 4 ) Note: Input Size: Õ B (N 3 ) D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

32 Remark All algorithms perform projection in both axes and solve the respective resultants. Hence, the following is crucial: Theorem (Solving Projections) The complexity for the solutions of the projections is: Õ B (n 12 + n 10 τ 2 ) = ÕB(N 12 ) since deg(res x ) = deg(res y ) = O(n 2 ) and L(res x ) = L(res y ) = O(nτ). The various methods differentiate on the matching procedure. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

34 GRID Algorithm 1 Project solutions on axes and find roots. 2 Generate all candidate pairs. 3 Check with BIVARIATESIGNAT for solutions. Theorem (GRID Complexity) Isolating all real roots of the system using GRID has complexity Õ B (n 14 + n 13 τ), provided that τ = O(n 3 ) D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

35 GRID Example Assume we want to solve the system: { f (x, y) = y x 3 + 2x 1 g(x, y) = y x 2 The system has solutions: (+0.45, +0.20) ( 1.25, +1.55) (+1.80, +3.25) D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

36 GRID Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

37 GRID Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

38 GRID Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

39 GRID Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

40 GRID Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

42 Generic Position Definition (Generic Position) Two polynomials f, g R[x, y] are in generic position if (f (ϱ, β 1 ) = g(ϱ, β 1 ) = 0) (f (ϱ, β 2 ) = g(ϱ, β 2 ) = 0) β 1 = β 2. Leading coefficients wrt y are not zero when specializing x. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

43 Solutions in Generic Position Theorem (Solution Existence) Let A, B square-free and co-prime polynomials in generic position. If SR j (x, y) = sr j (x)y j + sr j,j 1 (x)y j sr j,0 (x), then if (ϱ, β) is a real solution of the system A = B = 0, then there exists k, such that sr 0 (ϱ) =... = sr k 1(ϱ) = 0, sr k (ϱ) 0 and β = 1 k sr k,k 1 (ϱ) sr k (ϱ) What we need is: The minimum k such that sr k (ϱ) = psc k (ϱ) 0. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

44 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

45 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

46 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

47 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

48 How to Compute K? Naive: Evaluate all psc j (x) at x = ϱ and pick k = min j (psc j (ϱ)) 0. Better: Assume that it works for j = k > 0. It follows that psc j (ϱ) = 0 j {0, 1,..., k 1}. Hence ϱ appears on the gcd(psc 0,..., psc k 1 ). We define the polynomials: as well as Φ 0 (x) = psc 0 (x) gcd(psc 0 (x), psc 0 (x)) Φ j (x) = gcd(φ j 1 (x), psc j (x)) Γ j = Φ j 1(x), j {1,..., n 1} Φ j (x) k = j + 1 Complexity: Index k can be computed in time Õ B (n 8 τ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

49 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

50 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

51 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

52 Matching Solutions in M RUR - FIND. k computation yields first k such that psc k (x) 0 at x = ϱ. SR k (x = ϱ) (Z[ϱ])[y] has k roots. Generic Position At most 1 root. Hence 1 root with multiplicity k of the form: R(ϱ) = 1 k sr k,k 1 (ϱ) psc k (ϱ). Binary search for the unique interval such that: Q j < R(ϱ) < Q j+1 Complexity: The matching procedure takes time ÕB(n 7 τ) for each candidate on x-axis. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

53 Regarding Genericity But is the algorithm generic? YES! D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

54 Regarding Genericity But is the algorithm generic? YES! D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

55 Performing Shear. We are performing the transformation (X, Y ) (X + α Y, Y ) for some α Z. Complexity: The deterministic computation of α has complexity Õ B (n 9 τ). Bad practical performance. Random shear is the choice. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

56 M RUR Algorithm 1 Project solutions on axes and find roots. 2 Find Intermediate Points on y-axis. 3 For each x solution, compute k x. 4 Find suitable interval on y-axis to match solutions. Theorem (M RUR Complexity) Isolating all real roots of the system using M RUR has complexity Õ B (n 12 + n 10 τ 2 ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

57 M RUR Example Assume we want to solve the system: { f (x, y) = y x 3 + 2x 1 g(x, y) = y x 2 The system is not in generic position. The shear (x, y) (x + 3y, y) makes it generic: { f (x, y) = 27y 3 27xy 2 + (7 9x 2 )y x 3 + 2x 1 g(x, y) = 9y 2 + (1 6x)y x 2 The sheared system has solutions: ( 0.15, +0.20) ( 5.91, +1.55) ( 7.94, +3.25) D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

58 M RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

59 M RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

60 M RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

61 M RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

62 M RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

64 G RUR Algorithm 1 Project solutions on axes and find roots. 2 Find Intermediate Points on y-axis. 3 For each candidate solution ϱ on x-axis: Compute H(y) = gcd( f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Check for solutions of H(y) on candidate intervals along the y-axis. Theorem (G RUR Complexity) Isolating all real roots of the system using G RUR has complexity Õ B (n 12 + n 10 τ 2 ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

65 G RUR Example Assume we want to solve the system: { f (x, y) = y x 3 + 2x 1 g(x, y) = y x 2 The system has solutions: (+0.45, +0.20) ( 1.25, +1.55) (+1.80, +3.25) D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

66 G RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

67 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

68 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

69 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

70 G RUR Example Assume ϱ R alg such that res y (f, g)(ϱ) = 0. We want to compute H = gcd(f (ϱ, y), g(ϱ, y)) (Z[ϱ])[y]. Since res y (f, g)(α) = 0 we have ϱ 3 ϱ 2 2ϱ + 1 = 0. Or equivalently: ϱ 3 = ϱ 2 + 2ϱ 1 Hence f (ϱ, y) = y ϱ 3 + 2ϱ 1 = y ϱ 2 = g(ϱ, y). As a result gcd(f (ϱ, y), g(ϱ, y)) = y ϱ 2. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

71 G RUR Example The solutions of the system come from the H = gcd (Z[ϱ])[y]. Consequently, by substituting y with Intermediate Points H must change sign at x = ϱ. The Intermediate Points are: Q 1 = 1 Q 2 = Q 3 = Q 4 = 5 Checking interval [Q 1, Q 2 ] : { H1 (x) = 1 x 2 H 2 (x) = x 2 Note that H 1 (x) < 0 x R. Hence, we need to find a root ϱ R alg such that H 2 (ϱ) > 0. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

72 G RUR Example The solutions of the system come from the H = gcd (Z[ϱ])[y]. Consequently, by substituting y with Intermediate Points H must change sign at x = ϱ. The Intermediate Points are: Q 1 = 1 Q 2 = Q 3 = Q 4 = 5 Checking interval [Q 1, Q 2 ] : { H1 (x) = 1 x 2 H 2 (x) = x 2 Note that H 1 (x) < 0 x R. Hence, we need to find a root ϱ R alg such that H 2 (ϱ) > 0. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

73 G RUR Example The solutions of the system come from the H = gcd (Z[ϱ])[y]. Consequently, by substituting y with Intermediate Points H must change sign at x = ϱ. The Intermediate Points are: Q 1 = 1 Q 2 = Q 3 = Q 4 = 5 Checking interval [Q 1, Q 2 ] : { H1 (x) = 1 x 2 H 2 (x) = x 2 Note that H 1 (x) < 0 x R. Hence, we need to find a root ϱ R alg such that H 2 (ϱ) > 0. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

74 G RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

75 G RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

76 G RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

77 G RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

78 G RUR Example Working similarly... D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

79 G RUR Example D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

81 Applications µ simultaneous inequalities in two variables, can be solved in time: Õ B (µn 12 + µn 11 τ + n 10 τ 2 ). Topology of Real Plane Algebraic Curves. Theorem (Comlexity) We can compute the topology of a real plane algebraic curve, defined by a polynomial of degree n and bitsize τ, in time: Õ B (n 12 + n 10 τ 2 ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

82 Applications µ simultaneous inequalities in two variables, can be solved in time: Õ B (µn 12 + µn 11 τ + n 10 τ 2 ). Topology of Real Plane Algebraic Curves. Theorem (Comlexity) We can compute the topology of a real plane algebraic curve, defined by a polynomial of degree n and bitsize τ, in time: Õ B (n 12 + n 10 τ 2 ). D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

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

84 Library and Sample Usage MAPLE Code. Homepage of the library: stud1098/slv/ Sample Usage: f := 1 + 2*x + xˆ2*y - 5*x*y + xˆ2: g := 2*x + y - 3: bivsols := SLV:-solveGRID ( f, g ): SLV:-display_2 ( bivsols ); < 2*xˆ2-12*x+1, [ 3, 7], >, < xˆ2+6*x-25, [ -2263/256, -35/4], > < x-1, [ 1, 1], 1 >, < x-1, [ 1, 1], 1 > < 2*xˆ2-12*x+1, [3/64, 3/32], e-1 >, < xˆ2+6*x-25, [23179/8192, 2899/1024], > D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

85 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

86 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

87 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

88 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

89 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

90 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

91 GCD in Extension Field, Filtering and Speedup. GCD in Extension Field. [Hoeij and Monagan ] Filtering Interval and floating point arithmetic. Quadratic Interval Refinement [Abbot ] Iterations based on total degree of input polynomials. GCD computation. Finally exact algorithms and computation. M RUR pre-computation filtering. Speedup GRID and M RUR 10 G RUR 2 D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

92 Summary We propose three algorithms for real solving of bivariate polynomial systems. GRID ÕB(N 14 ). M RUR and G RUR ÕB(N 12 ). µ simultaneous inequalities in two variables are solved in Õ B (µn 12 ). The topology of a real plane algebraic curve can be computed in Õ B (N 12 ). Novel and robust MAPLE implementation. Outlook Express solutions of the sheared system in the original coordinate system. D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66

93 THANK YOU! D. I. Diochnos (Univ. of Athens, µ Q λ ) Real Solving on Bivariate Algebraic Systems Mar / 66