A Simple Quantifier-free Formula of Positive Semidefinite Cyclic - PowerPoint PPT Presentation

Research Background Research Result Examples Acknowledgements A Simple Quantifier-free Formula of Positive Semidefinite Cyclic Ternary Quartic Forms 1 Jingjun Han School of Mathematical Sciences, Peking University, Beijing, 100871, China October 27, 2012 1 hanjingjunfdfz@gmail.com Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements 1 Shing-Tung Yau High School Mathematics Award 2 Criterions on Equality of Symmetric Inequalities method 3 A quantifier-free formula of cyclic ternary quartic forms Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Background 1 Research Result 2 Main Result Preliminaries Lemma 6 Lemma 8 Theorem 9 Lemma 10 Examples 3 Acknowledgements 4 Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Background 1 A. Tarski, who gave a first quantifier elimination method for real closed fields in the 1930s Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Background 1 A. Tarski, who gave a first quantifier elimination method for real closed fields in the 1930s 2 G. E. Collins, who introduced a so-called cylindrical alge- braic decomposition (CAD) algorithm for QE problem in the 1970s Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Background 1 A. Tarski, who gave a first quantifier elimination method for real closed fields in the 1930s 2 G. E. Collins, who introduced a so-called cylindrical alge- braic decomposition (CAD) algorithm for QE problem in the 1970s 3 Many researchers have studied a special quantifier elimina- tion problem ( ∀ x ∈ R )( x 4 + px 2 + qx + r ≥ ( > )0) Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Background 1 A. Tarski, who gave a first quantifier elimination method for real closed fields in the 1930s 2 G. E. Collins, who introduced a so-called cylindrical alge- braic decomposition (CAD) algorithm for QE problem in the 1970s 3 Many researchers have studied a special quantifier elimina- tion problem ( ∀ x ∈ R )( x 4 + px 2 + qx + r ≥ ( > )0) 4 Gonz´ alez-Vega etc. and Yang etc. developed theory on root classification of polynomials independently, Yang named the theory complete discrimination systems . Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Background Choi etc, Harris, Timofte, Yao etc. have studied quantifier-free formula of symmetric forms. Choi etc.,1987 x i ∈ R + , i = 1 , 2 , . . . , n , the degree of form T ( x 1 , x 2 , . . . , x n ) equals three, then T ≥ 0 ⇐ ⇒ T (1 , 0 , 0 , . . . , 0) ≥ 0 T (1 , 1 , 0 , 0 , . . . , 0) ≥ 0 , . . . , T (1 , 1 , 1 , . . . , 1 , 0) ≥ 0 ; T (1 , 1 , 1 , . . . , 1) ≥ 0. Harris, 1999 x , y , z ∈ R + , the degree of form P ( x , y , z ) equals 4 or 5, then P ( x , y , z ) ≥ 0 ⇐ ⇒ P ( x , 1 , 1) ≥ 0 , P ( x , 1 , 0) ≥ 0. Yao etc., 2008 Give an algorithm to compute a quantifier free formula of the positive semidefinite symmetric forms of degree d with n variables in R n ( d ≤ 5, n is given.) Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements The author developed the so-called Criterions on Equality of Symmetric Inequalities method , which can prove these results. Question How about cyclic forms? Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Han, 2011 For cyclic form P with degree 3 of 3 variables, namely P ( a , b , c ) = m ( a 3 + b 3 + c 3 )+ n ( a 2 b + b 2 c + c 2 a )+ s ( ab 2 + bc 2 + ca 2 )+3 tabc ≥ 0 , ∀ a , b , c ∈ R + , P ≥ 0 ⇐ ⇒ P (1 , 1 , 1) ≥ 0 ; [ ∀ a ≥ 0 , P ( a , 1 , 0) ≥ 0] . Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements The author once studied cyclic ternary quartic forms, namely x 4 + k x 2 y 2 + l � � � x 2 yz ( ∀ x , y , z ∈ R )[ F ( x , y , z ) = cyc cyc cyc xy 3 ≥ 0] , � � x 3 y + n + m cyc cyc but not obtained a quantifier-free formula. Difficulty Two quantifiers (Due to homogenous). Difficult to obtain automatically by previous methods or quantifier elimi- nation tools. Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Research Method 1 Apply Criterions on Equality of Symmetric Inequalities method to reduce number of quantifiers to one 2 Apply the theory of complete discrimination systems and function RealTriangularize in Maple15 (or other tools) for solving the reduced case (QEPCAD, Discover). Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Main Result Preliminaries Lemma 6 Lemma 8 Theorem 9 Lemma 10 Research Result x 4 + k x 2 y 2 + l � � � x 2 yz + m � x 3 y + n � xy 3 , F ( x , y , z ) = cyc cyc cyc cyc cyc then ( ∀ x , y , z ∈ R ) [ F ( x , y , z ) ≥ 0] is equivalent to ∨ ( g 4 = 0 ∧ f 2 = 0 ∧ (( g 1 = 0 ∧ m ≥ 1 ∧ m ≤ 4) ∨ ( g 1 > 0 ∧ g 2 ≥ 0) ∨ ( g 1 > 0 ∧ g 3 ≥ 0))) ∨ ( g 2 4 + f 2 2 > 0 ∧ f 1 > 0 ∧ f 3 = 0 ∧ f 4 ≥ 0) ∨ ( g 2 4 + f 2 2 > 0 ∧ f 1 > 0 ∧ f 3 > 0 ∧ (( f 5 > 0 ∧ ( f 6 < 0 ∨ f 7 < 0)) ∨ ( f 5 = 0 ∧ f 7 < 0))) Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Main Result Preliminaries Lemma 6 Lemma 8 Theorem 9 Lemma 10 where f 1 :=2 + k − m − n , f 2 :=4 k + m + n − 8 − 2 l , f 3 :=1 + k + m + n + l , f 4 :=3(1 + k ) − m 2 − n 2 − mn , f 5 := − 4 k 3 m 2 − 4 k 3 n 2 − 4 k 2 lm 2 + 4 k 2 lmn − 4 k 2 ln 2 − kl 2 m 2 + 4 kl 2 mn − kl 2 n 2 + 8 klm 3 + 6 klm 2 n + 6 klmn 2 + 8 kln 3 − 2 km 4 + 10 km 3 n − 3 km 2 n 2 + 10 kmn 3 − 2 kn 4 + l 3 mn − 9 l 2 m 2 n − 9 l 2 mn 2 + lm 4 + 13 lm 3 n − 3 lm 2 n 2 + 13 lmn 3 + ln 4 − 7 m 5 − 8 m 4 n − 16 m 3 n 2 − 16 m 2 n 3 − 8 mn 4 Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Main Result Preliminaries Lemma 6 Lemma 8 Theorem 9 Lemma 10 − 7 n 5 + 16 k 4 + 16 k 3 l − 32 k 2 lm − 32 k 2 ln + 12 k 2 m 2 − 48 k 2 mn + 12 k 2 n 2 − 4 kl 3 + 4 kl 2 m + 4 kl 2 n − 12 klm 2 − 60 klmn − 12 kln 2 + 40 km 3 + 48 km 2 n + 48 kmn 2 + 40 kn 3 − l 4 + 10 l 3 m + 10 l 3 n − 21 l 2 m 2 + 12 l 2 mn − 21 l 2 n 2 + 10 lm 3 + 48 lm 2 n + 48 lmn 2 + 10 ln 3 − 17 m 4 − 14 m 3 n − 21 m 2 n 2 − 14 mn 3 − 17 n 4 − 16 k 3 + 32 k 2 l − 48 k 2 m − 48 k 2 n + 80 kl 2 − 48 klm − 48 kln + 96 km 2 + 48 kmn + 96 kn 2 − 24 l 3 − 24 l 2 m − 24 l 2 n + 24 lm 2 − 24 lmn + 24 ln 2 − 16 m 3 − 48 m 2 n − 48 mn 2 − 16 n 3 − 96 k 2 − 64 kl + 64 km + 64 kn + 96 l 2 − 32 lm − 32 ln − 16 m 2 − 32 mn − 16 n 2 + 64 k − 128 l + 64 m + 64 n + 128 , f 6 :=4 k 2 + 2 kl − 4 km − 4 kn + l 2 − 7 lm − 7 ln + 13 m 2 − mn + 13 n 2 − 40 k + 20 l + 8 m + 8 n − 32 , Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms

Research Background Research Result Examples Acknowledgements Main Result Preliminaries Lemma 6 Lemma 8 Theorem 9 Lemma 10 f 7 := − 768 + 352 k 2 − 332 l 2 + 180 n 2 + 180 m 2 + 56 k 3 − 8 k 4 + 14 l 3 + 132 n 3 + 132 m 3 + 42 n 4 + 42 m 4 − 480 k − 60 lmn − 192 n + 32 klmn − 192 m + 912 l + l 4 − 354 kmn + 158 kln + 158 klm + 26 k 2 mn − 11 kln 2 + 22 k 2 lm + 22 k 2 ln − 45 kmn 2 − 90 lm 2 n − 45 km 2 n − 11 klm 2 + 23 l 2 mn − 90 lmn 2 + kl 2 m + kl 2 n + 36 mn − 480 km + 592 kl − 480 kn − 60 lm − 60 ln + 8 k 3 m + 8 k 3 n − 20 k 2 l + 32 k 2 n + 32 k 2 m − 12 k 3 l + 234 mn 2 + 234 m 2 n − 192 ln 2 − 258 kn 2 − 192 lm 2 − 258 km 2 + 116 l 2 m + 116 l 2 n + 87 m 3 n + 87 mn 3 − 15 kn 3 + 90 m 2 n 2 − 30 ln 3 − 15 km 3 − 30 lm 3 + 25 l 2 m 2 + 25 l 2 n 2 − 14 k 2 m 2 − 14 k 2 n 2 − 146 kl 2 − 10 l 3 m − 10 l 3 n − 2 k 2 l 2 + 3 kl 3 , g 1 := k − 2 m + 2 , g 2 := 4 k − m 2 − 8 , g 3 := 8 + m − 2 k , g 4 = m − n . Jingjun Han A Quantifier-free Formula of Cyclic Ternary Quartic Forms