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Counting Signatures of Monic Polynomials Nomie Combe 1 & Vincent Jug 2 1: I2M (Aix-Marseille Universit & CNRS) 2: LSV (ENS Paris-Saclay & CNRS) 21/03/2017 following a work of Norbert ACampo N. Combe & V. Jug

  1. Counting Signatures: First Steps Evaluating s c , d , r “ # t signatures with parameters p c , d , r qu Recursion Formula for Facets ÿ s 0 , d ` 1 , 0 “ s 0 , d 1 , 0 ˆ s 0 , d 2 , 0 ˆ s 0 , d 3 , 0 ˆ s 0 , d 4 , 0 d 1 ` d 2 ` d 3 ` d 4 “ d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  2. Counting Signatures: First Steps Evaluating s c , d , r “ # t signatures with parameters p c , d , r qu Recursion Formula for Facets ÿ s 0 , d ` 1 , 0 “ s 0 , d 1 , 0 ˆ s 0 , d 2 , 0 ˆ s 0 , d 3 , 0 ˆ s 0 , d 4 , 0 d 1 ` d 2 ` d 3 ` d 4 “ d Proof: 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  3. Counting Signatures: First Steps Evaluating s c , d , r “ # t signatures with parameters p c , d , r qu Recursion Formula for Facets ÿ s 0 , d ` 1 , 0 “ s 0 , d 1 , 0 ˆ s 0 , d 2 , 0 ˆ s 0 , d 3 , 0 ˆ s 0 , d 4 , 0 d 1 ` d 2 ` d 3 ` d 4 “ d Proof: 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  4. Counting Signatures: First Steps Evaluating s c , d , r “ # t signatures with parameters p c , d , r qu Recursion Formula for Facets ÿ s 0 , d ` 1 , 0 “ s 0 , d 1 , 0 ˆ s 0 , d 2 , 0 ˆ s 0 , d 3 , 0 ˆ s 0 , d 4 , 0 d 1 ` d 2 ` d 3 ` d 4 “ d Proof: 6 3 7 3 0 4 1 4 5 3 2 4 4 3 3 4 3 3 H 2 3 1 3 0 2 0 3 3 2 2 2 1 2 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  5. Counting Signatures: First Steps Evaluating s c , d , r “ # t signatures with parameters p c , d , r qu Recursion Formula for Facets ÿ s 0 , d ` 1 , 0 “ s 0 , d 1 , 0 ˆ s 0 , d 2 , 0 ˆ s 0 , d 3 , 0 ˆ s 0 , d 4 , 0 d 1 ` d 2 ` d 3 ` d 4 “ d Counting Facets with Fuss-Catalan Numbers (A’Campo 17) 1 ˆ 4 d ˙ s 0 , d , 0 “ 3 d ` 1 d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  6. Counting Signatures: First Steps Evaluating s c , d , r “ # t signatures with parameters p c , d , r qu Recursion Formula for Facets ÿ s 0 , d ` 1 , 0 “ s 0 , d 1 , 0 ˆ s 0 , d 2 , 0 ˆ s 0 , d 3 , 0 ˆ s 0 , d 4 , 0 d 1 ` d 2 ` d 3 ` d 4 “ d Counting Facets with Fuss-Catalan Numbers (A’Campo 17) 1 ˆ 4 d ˙ s 0 , d , 0 “ 3 d ` 1 d ñ What next? N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  7. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  8. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  9. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  10. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  11. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  12. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  13. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  14. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  15. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 3 Recursive decomposition 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  16. Counting Signatures: Some Tools Strategy: Use recursion formulæ and generating functions 1 Generating function S p x , y , z q “ ř s c , d , r x c y d z r 2 Canonical splitting 3 Recursive decomposition 6 3 7 3 0 4 1 4 5 3 2 4 4 3 3 4 3 3 H 2 3 1 3 0 2 0 3 3 2 2 2 1 2 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  17. Counting Signatures: Some More Tools 1 Variant of signatures: contact signatures 18 19 20 21 22 23 24 17 25 16 26 15 27 14 13 12 0 11 1 10 2 9 8 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  18. Counting Signatures: Some More Tools 1 Variant of signatures: contact signatures 18 19 20 21 22 23 24 17 25 16 26 15 27 14 28 13 ´ 1 12 0 11 1 10 2 9 8 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  19. Counting Signatures: Some More Tools 1 Variant of signatures: contact signatures with generating series C p x , y , z q 18 19 20 21 22 23 24 17 25 16 26 15 27 14 28 13 ´ 1 12 0 11 1 10 2 9 8 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  20. Counting Signatures: Some More Tools 1 Variant of signatures: contact signatures with generating series C p x , y , z q 2 Another variant of signatures with generating series D p x , y , z q 18 19 20 21 22 23 24 17 25 16 26 15 27 14 28 13 ´ 1 12 0 11 1 10 2 9 8 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  21. Counting Signatures: The End is Near Three algebraic equations (using bijective proofs) S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q C “ DS D “ 1 ` xy C 4 D 2 {p 1 ´ x 2 y C 4 D q N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  22. Counting Signatures: The End is Near Three algebraic equations (using bijective proofs) S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q C “ DS D “ 1 ` xy C 4 D 2 {p 1 ´ x 2 y C 4 D q Theorem The generating function S p x , y , z q is algebraic! N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  23. Counting Signatures: The End is Near Three algebraic equations (using bijective proofs) S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q C “ DS D “ 1 ` xy C 4 D 2 {p 1 ´ x 2 y C 4 D q Theorem The generating function S p x , y , z q is algebraic! and its minimal polynomial is not so nice. . . x 4 p x ` 1 q 4 ` S 4 yv x 4 ´ x 8 ` 4 vx p x 4 ` x 2 v ` v 2 q ´ p x 2 ` v 2 q 2 ` S 4 yv 5 ˘ ` “ 0 with v “ x 2 z ` 1 {p S ´ 1 q . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  24. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  25. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � 2 Finding a linear DE satisfied by S Size overflow � N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  26. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � 2 Finding a linear DE satisfied by S Size overflow � 3 Using the 3 equations! Makes the job � S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q C “ DS D “ 1 ` xy C 4 D 2 {p 1 ´ x 2 y C 4 D q N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  27. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � 2 Finding a linear DE satisfied by S Size overflow � 3 Using the 3 equations! Makes the job � S “ 1 ` y C 4 ´ x 2 yz C 4 ` x 2 yz SC 4 C “ DS D “ 1 ` xy C 4 D 2 ´ x 2 y C 4 D ` x 2 y C 4 D 2 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  28. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � 2 Finding a linear DE satisfied by S Size overflow � 3 Using the 3 equations! Makes the job � S “ 1 ` y C 4 ´ x 2 yz C 4 ` x 2 yz SC 4 C “ DS D “ 1 ` xy C 4 D 2 ´ x 2 y C 4 D ` x 2 y C 4 D 2 Two more lemmas: s c , d , r ą 0 ñ 2 r ď c ď 2 d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  29. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � 2 Finding a linear DE satisfied by S Size overflow � 3 Using the 3 equations! Makes the job � S “ 1 ` y C 4 ´ x 2 yz C 4 ` x 2 yz SC 4 C “ DS D “ 1 ` xy C 4 D 2 ´ x 2 y C 4 D ` x 2 y C 4 D 2 Two more lemmas: s c , d , r ą 0 ñ 2 r ď c ď 2 d and s c , d , r ď 30 d ` 1 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  30. Counting Signatures Efficiently Three ideas for computing s c , d , r 1 Using directly the minimal polynomial of S Did not work � 2 Finding a linear DE satisfied by S Size overflow � 3 Using the 3 equations! Makes the job � S “ 1 ` y C 4 ´ x 2 yz C 4 ` x 2 yz SC 4 C “ DS D “ 1 ` xy C 4 D 2 ´ x 2 y C 4 D ` x 2 y C 4 D 2 Two more lemmas: s c , d , r ą 0 ñ 2 r ď c ď 2 d and s c , d , r ď 30 d ` 1 Corollary The family of coefficients p s c , d , r q c ď C , d ď D , r ď R can be computed in time O p min t C , D , R u 2 min t C , D u 2 D 4 q . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  31. Contents Signatures of Monic Polynomials 1 Counting Signatures 2 Asymptotic Estimations 3 Conclusion 4 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  32. What if d Ñ `8 ? Problem: Fix c and r and evaluate lim s c , d , r when d Ñ `8 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  33. What if d Ñ `8 ? Problem: Fix c and r and evaluate lim s c , d , r when d Ñ `8 Two ideas: 1 Singularity analysis of S Difficult � (Several branches in multivariate environment) N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  34. What if d Ñ `8 ? Problem: Fix c and r and evaluate lim s c , d , r when d Ñ `8 Two ideas: 1 Singularity analysis of S Difficult � 2 Study a class of typical signatures! Successful � N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  35. What if d Ñ `8 ? Problem: Fix c and r and evaluate lim s c , d , r when d Ñ `8 Two ideas: 1 Singularity analysis of S Difficult � 2 Study a class of typical signatures! Successful � (Each component has at most 1 contact point, of small valency) 10 9 8 13 14 15 16 17 13 14 15 16 17 12 18 12 18 7 11 11 19 11 19 ✗ ✗ ✗ 6 0 10 0 10 0 9 1 9 1 5 1 8 2 8 2 4 3 2 7 6 5 4 3 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  36. What if d Ñ `8 ? Problem: Fix c and r and evaluate lim s c , d , r when d Ñ `8 Two ideas: 1 Singularity analysis of S Difficult � 2 Study a class of typical signatures! Successful � (Each component has at most 1 contact point, of small valency) 13 14 15 16 17 13 14 15 16 17 12 18 12 18 11 19 11 19 � � 10 0 10 0 9 1 9 1 8 2 8 2 7 6 5 4 3 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  37. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  38. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: H N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  39. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: 3 4 0 2 1 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  40. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: 5 6 7 4 8 3 0 2 1 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  41. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: 7 8 0 6 5 2 1 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  42. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: 7 8 0 1 6 2 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  43. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: 5 6 3 4 7 8 0 2 1 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  44. Counting Typical Signatures (1/2) Another generating function: T p x , y , z q “ ř t c , d , r x c y d z r Lemma #4 T “ 1 ` y T 4 ` 4 xy 2 T 8 ` x 2 y 2 z T 8 . Proof: 6 7 5 8 0 4 1 3 2 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  45. Counting Typical Signatures (2/2) Algebraic equation with triangular system of variables T “ 1 ` y T 4 ` 4 x y 2 T 8 ` x 2 y 2 z T 8 . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  46. Lagrange Inversion (with 3 variables) Algebraic equation with triangular system of variables T “ 1 ` y T 4 ` 4 x y 2 T 8 ` x 2 y 2 z T 8 . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  47. Lagrange Inversion (with 3 variables) Algebraic equation with triangular system of variables T “ 1 ` y T 4 ` 4 x y 2 T 8 ` x 2 y 2 z T 8 . Exact and asymptotic evaluations t c , d , r “ 1 c ě 2 r ¨ 1 d ě 2 c ´ 2 r ¨ 4 c ´ 2 r ˆ ˙ 4 d c ` 3 d ´ r ` 1 c ´ 2 r , d ´ 2 c ´ 2 r , r , c ` 3 d ´ r c 16 r ¨ 4 4 d 27 π ¨ 4 c 3 c ¨ 3 r 1 c ě 2 r 2 3 3 d ¨ d c ´ r ´ 3 { 2 t c , d , r „ r ! p c ´ 2 r q ! ¨ N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  48. 18 19 20 21 22 23 24 Typical Signatures are Typical 17 16 25 Main tool: Reducing a signature 15 26 14 27 13 12 0 11 1 10 2 9 8 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  49. 18 19 20 21 22 23 24 Typical Signatures are Typical 17 16 25 Main tool: Reducing a signature 15 26 14 27 13 12 0 11 1 10 2 9 8 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  50. 14 1 15 1 16 1 17 1 18 1 19 1 20 1 Typical Signatures are Typical 13 1 12 1 21 1 Main tool: Reducing a signature 11 1 22 1 10 1 23 1 9 1 8 1 0 1 1 1 2 1 7 1 6 1 5 1 4 1 3 1 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  51. 14 1 15 1 16 1 17 1 18 1 19 1 20 1 Typical Signatures are Typical 13 1 12 1 21 1 Main tool: Reducing a signature 11 1 22 1 10 1 23 1 9 1 8 1 0 1 1 1 2 1 7 1 6 1 5 1 4 1 3 1 Bounding lemma ´ ¯ d c ´ r ´ 5 { 2 ¨ 4 4 d At most O signatures reduce to a non-typical signature σ . 3 3 d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  52. 14 1 15 1 16 1 17 1 18 1 19 1 20 1 Typical Signatures are Typical 13 1 12 1 21 1 Main tool: Reducing a signature 11 1 22 1 10 1 23 1 Proof: with C components: 9 1 Fill the regions (at most 8 c ) 8 1 0 1 1 1 2 1 7 1 6 1 5 1 4 1 3 1 Bounding lemma ´ ¯ d c ´ r ´ 5 { 2 ¨ 4 4 d At most O signatures reduce to a non-typical signature σ . 3 3 d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  53. ? Typical Signatures are Typical Main tool: Reducing a signature Proof: with C components: Fill the regions (at most 8 c ) ? ? Place the C components Bounding lemma ´ ¯ d c ´ r ´ 5 { 2 ¨ 4 4 d At most O signatures reduce to a non-typical signature σ . 3 3 d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  54. ? Typical Signatures are Typical Main tool: Reducing a signature Proof: with C components: Fill the regions (at most 8 c ) ? ? Place the C components Non-typical ô C ď c ´ r ´ 1 Bounding lemma ´ ¯ d c ´ r ´ 5 { 2 ¨ 4 4 d At most O signatures reduce to a non-typical signature σ . 3 3 d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  55. ? Typical Signatures are Typical Main tool: Reducing a signature Proof: with C components: Fill the regions (at most 8 c ) ? ? Place the C components Non-typical ô C ď c ´ r ´ 1 Fixing c ñ finite number of reductions Bounding lemma ´ ¯ d c ´ r ´ 5 { 2 ¨ 4 4 d At most O signatures reduce to a non-typical signature σ . 3 3 d Theorem c 27 π ¨ 4 c 3 c ¨ 3 r 16 r ¨ 4 4 d 1 c ě 2 r 2 3 3 d ¨ d c ´ r ´ 3 { 2 s c , d , r „ t c , d , r „ r ! p c ´ 2 r q ! ¨ N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  56. Contents Signatures of Monic Polynomials 1 Counting Signatures 2 Asymptotic Estimations 3 Conclusion 4 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  57. Conclusion We still need to. . . 1 Look for more efficient algorithms or closed-form formulæ N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  58. Conclusion We still need to. . . 1 Look for more efficient algorithms or closed-form formulæ 2 Study the distribution of c , r and p c , r q for large values of d N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  59. Conclusion We still need to. . . 1 Look for more efficient algorithms or closed-form formulæ 2 Study the distribution of c , r and p c , r q for large values of d 3 Ask you for other ideas and Thank you! N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  60. Counting Signatures: Three Lemmas (1/3) Lemma #1 S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  61. Counting Signatures: Three Lemmas (1/3) Lemma #1 S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q . Proof: 13 14 15 16 17 12 18 11 19 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  62. Counting Signatures: Three Lemmas (1/3) Lemma #1 S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q . Proof: 13 14 15 16 17 12 18 11 19 ´ 1 1 10 0 0 1 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  63. Counting Signatures: Three Lemmas (1/3) Lemma #1 S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q . Proof: 13 14 15 16 17 12 18 11 19 10 0 ´ 1 2 9 1 0 2 8 2 4 2 3 2 2 2 1 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  64. Counting Signatures: Three Lemmas (1/3) Lemma #1 S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q . Proof: 6 3 7 3 8 3 13 14 15 16 17 5 3 12 18 4 3 11 19 3 3 10 0 2 3 9 1 1 3 8 2 0 3 ´ 1 3 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  65. Counting Signatures: Three Lemmas (1/3) Lemma #1 S “ 1 ` y C 4 {p 1 ´ x 2 yz C 4 q . Proof: ´ 1 4 0 4 1 4 13 14 15 16 17 2 4 12 18 3 4 11 19 4 4 10 0 9 1 8 2 7 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  66. Counting Signatures: Three Lemmas (2/3) Lemma #2 C “ DS . N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  67. Counting Signatures: Three Lemmas (2/3) Lemma #2 C “ DS . Proof: 12 13 14 15 11 16 10 ´ 1 9 0 8 1 7 2 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

  68. Counting Signatures: Three Lemmas (2/3) Lemma #2 C “ DS . Proof: 12 13 14 15 8 1 11 16 ´ 1 1 10 ´ 1 0 1 9 0 1 1 8 1 7 1 2 1 7 2 6 1 5 1 4 1 3 1 6 5 4 3 N. Combe & V. Jugé Counting Signatures of Monic Polynomials

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