We have φ A ( x ) = φ ∗ 1 A ( x ) F A ( x ) = in F 2 [ x ] . (1) 1 + x D A ( x ) = � r If φ ∗ i =1 x b i , where 0 = b 1 < · · · < b r = D − max A , then f A ( n ) ≡ 1 mod 2 ⇐ ⇒ n ≡ b i mod D for some i . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
We have φ A ( x ) = φ ∗ 1 A ( x ) F A ( x ) = in F 2 [ x ] . (1) 1 + x D A ( x ) = � r If φ ∗ i =1 x b i , where 0 = b 1 < · · · < b r = D − max A , then f A ( n ) ≡ 1 mod 2 ⇐ ⇒ n ≡ b i mod D for some i . In any block of D consecutive integers, # { n : f A ( n ) is odd } = ℓ ( φ ∗ A ) = β 1 ( φ A ) # { n : f A ( n ) is even } = D − ℓ ( φ ∗ A ) = β 0 ( φ A ) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
In Reciprocals of Binary Power Series , which appeared in International Journal of Number Theory in 2006, Cooper, Eichhorn, and O’Bryant considered the fraction ℓ ( φ ∗ A ) / D , as we did in our paper. Here I instead consider the ordered pair β ( φ A ) := ( β 1 ( φ A ) , β 0 ( φ A )) , which gives more detailed information than reduced fractions. The first coordinate represents the number of times f A ( n ) is odd in a minimal period, and the second coordinate represents the number of times f A ( n ) is even in a minimal period. Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Robust polynomials Cooper, Eichhorn, and O’Bryant showed by direct computation that β 1 ( f ) ≤ β 0 ( f ) + 1 when deg( f ) < 8 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Robust polynomials Cooper, Eichhorn, and O’Bryant showed by direct computation that β 1 ( f ) ≤ β 0 ( f ) + 1 when deg( f ) < 8 . We call a polynomial f ( x ) robust if β 1 ( f ) > β 0 ( f ) + 1. This is equivalent to saying that β 1 ( f ) > ( D + 1) / 2, where D is the order of f ( x ). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
They also posed the problem of describing the set � β 1 ( f ) � β 0 ( f ) + β 1 ( f ) : f ( x ) is a polynomial . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
They also posed the problem of describing the set � β 1 ( f ) � β 0 ( f ) + β 1 ( f ) : f ( x ) is a polynomial . Since f ( x ) = 1 + x D has order D and β 1 ( f ) = ℓ ( f ∗ ( x )) = 1, we see the greatest lower bound of the set is 0. I will exhibit four sequences { f n } of polynomials such that β 1 ( f n ) − β 0 ( f n ) → ∞ , and, moreover, β 1 ( f n ) lim β 0 ( f n ) + β 1 ( f n ) = 1 . n →∞ Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
For n with standard binary representation n = 2 b k + 2 b k − 1 + · · · + 2 b 1 + 2 b 0 , define P n ( x ) = x b k + x b k − 1 + · · · + x b 1 + x b 0 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
For n with standard binary representation n = 2 b k + 2 b k − 1 + · · · + 2 b 1 + 2 b 0 , define P n ( x ) = x b k + x b k − 1 + · · · + x b 1 + x b 0 . For example, 11 = 2 3 + 2 1 + 2 0 , so P 11 ( x ) = x 3 + x + 1. Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
For n with standard binary representation n = 2 b k + 2 b k − 1 + · · · + 2 b 1 + 2 b 0 , define P n ( x ) = x b k + x b k − 1 + · · · + x b 1 + x b 0 . For example, 11 = 2 3 + 2 1 + 2 0 , so P 11 ( x ) = x 3 + x + 1. For odd n , consider the fraction ℓ ( P ∗ n ) ord( P n ) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Reciprocal Polynomials Definition For a polynomial f ( x ) of degree n , the reciprocal polynomial of f ( x ) is f ( R ) ( x ) := x n f (1 / x ). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Reciprocal Polynomials Definition For a polynomial f ( x ) of degree n , the reciprocal polynomial of f ( x ) is f ( R ) ( x ) := x n f (1 / x ). � � If order( f ( x )) = D , then order f ( R ) ( x ) = D . Thus � � β ( f ( x )) = β f ( R ) ( x ) , and the robustness of f ( x ) is equivalent to the robustness of f ( R ) ( x ). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Reciprocal Polynomials Definition For a polynomial f ( x ) of degree n , the reciprocal polynomial of f ( x ) is f ( R ) ( x ) := x n f (1 / x ). � � If order( f ( x )) = D , then order f ( R ) ( x ) = D . Thus � � β ( f ( x )) = β f ( R ) ( x ) , and the robustness of f ( x ) is equivalent to the robustness of f ( R ) ( x ). With A = { 0 = a 0 < a 1 < · · · < a j } , define ˜ A = { 0 , a j − a j − 1 , · · · , a j − a 1 , a j } . Then φ A , ( R ) ( x ) = φ ˜ A . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
First Theorem Theorem Fix r ≥ 3 . (i) The order of f r , 1 ( x ) := (1 + x )(1 + x 2 r − 1 + x 2 r ) divides 4 r − 1 . (ii) β 1 ( f r , 1 ) = 4 r − 3 r (iii) Hence β ( f r , 1 ) = (4 r − 3 r , 3 r − 1) and f r , 1 ( x ) is robust. Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 1 ( x ) = 1 + x + x 7 + x 9 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 1 ( x ) = 1 + x + x 7 + x 9 . ◮ order ( f 3 , 1 ( x )) = 4 3 − 1 = 63 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 1 ( x ) = 1 + x + x 7 + x 9 . ◮ order ( f 3 , 1 ( x )) = 4 3 − 1 = 63 ◮ β 1 ( f 3 , 1 ) = 4 3 − 3 3 = 37 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 1 ( x ) = 1 + x + x 7 + x 9 . ◮ order ( f 3 , 1 ( x )) = 4 3 − 1 = 63 ◮ β 1 ( f 3 , 1 ) = 4 3 − 3 3 = 37 ◮ β ( f 3 , 1 ) = (37 , 26) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 1 ( x ) = 1 + x + x 7 + x 9 . ◮ order ( f 3 , 1 ( x )) = 4 3 − 1 = 63 ◮ β 1 ( f 3 , 1 ) = 4 3 − 3 3 = 37 ◮ β ( f 3 , 1 ) = (37 , 26) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Proof Define r − 1 1 + x (2 r − 1)2 j + x 2 r 2 j � + x 4 r − 2 r . � � g r , 1 ( x ) = j =0 By a lemma, 1 + x 2 r − 1 + x 2 r � g r , 1 ( x ) = 1 + x 4 r − 1 . � Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Because r − 1 � g r , 1 (1) = (1 + 1 + 1) + 1 ≡ 0 (mod 2) , j =0 we know (1 + x ) | g r , 1 ( x ). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Because r − 1 � g r , 1 (1) = (1 + 1 + 1) + 1 ≡ 0 (mod 2) , j =0 we know (1 + x ) | g r , 1 ( x ). Write (1 + x ) h r , 1 ( x ) = g r , 1 ( x ), so 1 + x 2 r − 1 + x 2 r � (1 + x ) h r , 1 ( x ) = 1 + x 4 r − 1 . � Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Because r − 1 � g r , 1 (1) = (1 + 1 + 1) + 1 ≡ 0 (mod 2) , j =0 we know (1 + x ) | g r , 1 ( x ). Write (1 + x ) h r , 1 ( x ) = g r , 1 ( x ), so 1 + x 2 r − 1 + x 2 r � (1 + x ) h r , 1 ( x ) = 1 + x 4 r − 1 . � 1 + x 4 r − 1 � � Thus f r , 1 ( x ) | and f r , 1 h r , 1 = 1 + x 4 r − 1 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Rewrite r − 1 1 + x (2 r − 1)2 j + x 2 r 2 j � � � + x 4 r − 2 r g r , 1 ( x ) = j =0 to obtain r − 1 + x 4 r − 2 r . � � � 1 + x (2 r − 1)2 j (1 + x 2 j ) g r , 1 ( x ) = j =0 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Expand the product and rewrite, using 1 + x 2 j = (1 + x ) 2 j , to obtain 2 r − 1 g r , 1 ( x ) = 1 + x 4 r − 2 r + � x (2 r − 1) n (1 + x ) n n =1 2 r − 1 � � 1 + x 4 r − 2 r x (2 r − 1) n (1 + x ) n − 1 � = (1 + x ) + 1 + x n =1 4 r − 2 r − 1 2 r − 1 x j + � � x (2 r − 1) n (1 + x ) n − 1 . = (1 + x ) j =0 n =1 Ultimately, ( β 1 ( f r , 1 ) , β 0 ( f r , 1 )) = (4 r − 3 r , 3 r − 1). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Corollary The reciprocal polynomials f ( R ) , r , 1 = (1 + x )(1 + x + x 2 r ) are also robust with order dividing 4 r − 1 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Corollary The reciprocal polynomials f ( R ) , r , 1 = (1 + x )(1 + x + x 2 r ) are also robust with order dividing 4 r − 1 . Example Consider f ( R ) , 3 , 1 ( x ) = 1 + x 2 + x 8 + x 9 . ◮ order f ( R ) , 3 , 1 = 4 3 − 1 = 63 ◮ β � � = (37 , 26) f ( R ) , 3 , 1 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Theorem Fix r ≥ 3 . (i) The order of f r , 2 ( x ) := (1 + x )(1 + x 2 r + x 2 r +1 ) divides 4 r + 2 r + 1 . (ii) β 1 ( f r , 2 ) = 4 r − 3 r + 2 r (iii) β ( f r , 2 ) = (4 r − 3 r + 2 r , 3 r + 1) and f r , 2 ( x ) is robust. Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 2 ( x ) = 1 + x + x 8 + x 10 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 2 ( x ) = 1 + x + x 8 + x 10 . ◮ order ( f 3 , 2 ( x )) = 4 3 + 2 3 + 1 = 73 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 2 ( x ) = 1 + x + x 8 + x 10 . ◮ order ( f 3 , 2 ( x )) = 4 3 + 2 3 + 1 = 73 ◮ β 1 ( f 3 , 2 ) = 4 3 − 3 3 + 2 3 = 45 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 2 ( x ) = 1 + x + x 8 + x 10 . ◮ order ( f 3 , 2 ( x )) = 4 3 + 2 3 + 1 = 73 ◮ β 1 ( f 3 , 2 ) = 4 3 − 3 3 + 2 3 = 45 ◮ β ( f 3 , 2 ) = (45 , 28) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Consider f 3 , 2 ( x ) = 1 + x + x 8 + x 10 . ◮ order ( f 3 , 2 ( x )) = 4 3 + 2 3 + 1 = 73 ◮ β 1 ( f 3 , 2 ) = 4 3 − 3 3 + 2 3 = 45 ◮ β ( f 3 , 2 ) = (45 , 28) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Corollary The reciprocal polynomials f ( R ) , r , 2 ( x ) = (1 + x )(1 + x + x 2 r +1 ) are also robust with order dividing 4 r + 2 r + 1 . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Corollary The reciprocal polynomials f ( R ) , r , 2 ( x ) = (1 + x )(1 + x + x 2 r +1 ) are also robust with order dividing 4 r + 2 r + 1 . Example Consider f ( R ) , 3 , 2 ( x ) = 1 + x 2 + x 9 + x 10 . ◮ order f ( R ) , 3 , 2 = 73 ◮ β � � = (45 , 28) f ( R ) , 3 , 2 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Future Research Ideas ◮ Finding more families of robust polynomials ◮ Determining the cluster points of � � β 1 ( f ) β 0 ( f ) + β 1 ( f ) : f ( x ) is a polynomial ◮ Exploring properties of f A ( n ) in bases other than 2 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Acknowledgements ◮ The presenter acknowledges support from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students”. ◮ The presenter also wishes to thank Professor Bruce Reznick for his time, ideas, and encouragement. Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Recall f A ( n ) is the number of ways to write ∞ � ǫ i 2 i , where ǫ i ∈ A := { 0 = a 0 < a 1 < · · · < a z } . n = i =0 Expanding the sum, we see that n = ǫ 0 + ǫ 1 2 + ǫ 2 2 2 + · · · = ǫ 0 + 2 ( ǫ 1 + ǫ 2 2 + · · · ) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Recall f A ( n ) is the number of ways to write ∞ � ǫ i 2 i , where ǫ i ∈ A := { 0 = a 0 < a 1 < · · · < a z } . n = i =0 Expanding the sum, we see that n = ǫ 0 + ǫ 1 2 + ǫ 2 2 2 + · · · = ǫ 0 + 2 ( ǫ 1 + ǫ 2 2 + · · · ) We will now examine the asymptotic behavior of 2 r +1 − 1 � f A ( n ) . n =2 r Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Write A = { 0 = 2 b 1 , 2 b 2 , . . . , 2 b s , 2 c 1 + 1 , . . . , 2 c t + 1 } . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Write A = { 0 = 2 b 1 , 2 b 2 , . . . , 2 b s , 2 c 1 + 1 , . . . , 2 c t + 1 } . If n is even, then ǫ 0 = 0 , 2 b 2 , 2 b 3 , . . . , or 2 b s and � n − 2 b 2 � � n − 2 b 3 � � n − 2 b s � � n � f A ( n ) = f A + f A + f A + · · · + f A . 2 2 2 2 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Write A = { 0 = 2 b 1 , 2 b 2 , . . . , 2 b s , 2 c 1 + 1 , . . . , 2 c t + 1 } . If n is even, then ǫ 0 = 0 , 2 b 2 , 2 b 3 , . . . , or 2 b s and � n − 2 b 2 � � n − 2 b 3 � � n − 2 b s � � n � f A ( n ) = f A + f A + f A + · · · + f A . 2 2 2 2 Writing n = 2 ℓ , we have f A (2 ℓ ) = f A ( ℓ ) + f A ( ℓ − b 2 ) + f A ( ℓ − b 3 ) + · · · + f A ( ℓ − b s ) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
If n is odd, then ǫ 0 = 2 c 1 + 1 , 2 c 2 + 1 , . . . , or 2 c t + 1, and � n − (2 c 1 + 1) � � n − (2 c 2 + 1) � f A ( n ) = f A + f A 2 2 � n − (2 c t + 1) � + · · · + f A . 2 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
If n is odd, then ǫ 0 = 2 c 1 + 1 , 2 c 2 + 1 , . . . , or 2 c t + 1, and � n − (2 c 1 + 1) � � n − (2 c 2 + 1) � f A ( n ) = f A + f A 2 2 � n − (2 c t + 1) � + · · · + f A . 2 Writing n = 2 ℓ + 1, we have f A (2 ℓ + 1) = f A ( ℓ − c 1 ) + f A ( ℓ − c 2 ) + · · · + f A ( ℓ − c t ) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example If A = { 0 , 1 , 4 , 9 } = { 2 · 0 , 2 · 0 + 1 , 2 · 2 , 2 · 4 + 1 } , then we have f A (2 ℓ ) = f A ( ℓ ) + f A ( ℓ − 2) and f A (2 ℓ + 1) = f A ( ℓ ) + f A ( ℓ − 4) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
For positive integers k , m , and a z , let f A (2 k m ) f A (2 k m − 1) ω k ( m ) = . . . . f A (2 k m − a z ) We will show that for a z sufficiently large, there exists a fixed ( a z + 1) × ( a z + 1) matrix M such that for any k ≥ 0, ω k +1 = M ω k . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example Let A = { 0 , 1 , 3 , 4 } . Then f A (2 ℓ ) = f A ( ℓ ) + f A ( ℓ − 2) and f A (2 ℓ + 1) = f A ( ℓ ) + f A ( ℓ − 1) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
{ 0 , 1 , 3 , 4 } continued f A (2 k +1 m ) f A (2 k m ) + f A (2 k m − 2) f A (2 k +1 m − 1) f A (2 k m − 1) + f A (2 k m − 2) f A (2 k +1 m − 2) f A (2 k m − 1) + f A (2 k m − 3) ω k +1 ( m ) = = . f A (2 k +1 m − 3) f A (2 k m − 2) + f A (2 k m − 3) f A (2 k +1 m − 4) f A (2 k m − 2) + f A (2 k m − 4) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
{ 0 , 1 , 3 , 4 } continued f A (2 k +1 m ) f A (2 k m ) + f A (2 k m − 2) f A (2 k +1 m − 1) f A (2 k m − 1) + f A (2 k m − 2) f A (2 k +1 m − 2) f A (2 k m − 1) + f A (2 k m − 3) ω k +1 ( m ) = = . f A (2 k +1 m − 3) f A (2 k m − 2) + f A (2 k m − 3) f A (2 k +1 m − 4) f A (2 k m − 2) + f A (2 k m − 4) 1 0 1 0 0 0 1 1 0 0 and M = 0 1 0 1 0 satisfies ω k +1 ( m ) = M ω k ( m ). 0 0 1 1 0 0 0 1 0 1 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Theorem Let A , f A ( n ) , M, and ω k ( m ) be as above, with the additional assumption that there exists some odd a i ∈ A . Define 2 r +1 − 1 � s A ( r ) = f A ( n ) . n =2 r Let |A| denote the number of elements in the set A . Then s A ( r ) lim |A| r = c ( A ) , r →∞ where c ( A ) ∈ Q , so s A ( r ) ≈ c ( A ) |A| r . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example: A = { 0 , 2 , 3 } f A (2 ℓ ) = f A ( ℓ ) + f A ( ℓ − 1) f A (2 ℓ + 1) = f A ( ℓ − 1) f A (2 k +1 m ) f A (2 k m ) 1 1 0 f A (2 k +1 m − 1) = f A (2 k m − 1) 0 0 1 f A (2 k +1 m − 2) f A (2 k m − 2) 0 1 1 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Example: A = { 0 , 2 , 3 } f A (2 ℓ ) = f A ( ℓ ) + f A ( ℓ − 1) f A (2 ℓ + 1) = f A ( ℓ − 1) f A (2 k +1 m ) f A (2 k m ) 1 1 0 f A (2 k +1 m − 1) = f A (2 k m − 1) 0 0 1 f A (2 k +1 m − 2) f A (2 k m − 2) 0 1 1 The characteristic polynomial of M is g ( x ) = − ( x − 1)( x 2 − x − 1). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
{ 0 , 2 , 3 } continued 2 r +1 − 1 � s A ( r ) = f A ( n ) n =2 r 2 r − 1 � = ( f A (2 n ) + f A (2 n + 1)) n =2 r − 1 2 r − 1 � = ( f A ( n ) + f A ( n − 1) + f A ( n − 1)) n =2 r − 1 2 r − 1 � = s A ( r − 1) + 2 f A ( n − 1) n =2 r − 1 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
2 r − 1 f A ( n ) + 2 f A (2 r − 1 − 1) − 2 f A (2 r − 1) � s A ( r ) = s A ( r − 1) + 2 n =2 r − 1 = 3 s A ( r − 1) + 2 f A (2 r − 1 − 1) − 2 f A (2 r − 1) = 3 s A ( r − 1) + 2 F r − 2 − 2 F r − 1 = 3 s A ( r − 1) − 2 F r − 3 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
◮ Solution to homogeneous recurrence relation s A ( r ) = c 1 3 r ◮ Solution to inhomogeneous recurrence relation s A ( r ) = c 1 3 r + c 2 φ r + c 3 ¯ φ r + c 4 (1) r Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
s A ( r + 2) − s A ( r + 1) − s A ( r ) = c 1 3 r (3 2 − 3 − 1) + c 2 φ r ( φ 2 − φ − 1) φ 2 − ¯ φ − 1) + c 4 (1 2 − 1 − 1) + c 3 ¯ φ r (¯ = c 1 3 r · 5 − c 4 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
s A ( r + 2) − s A ( r + 1) − s A ( r ) = c 1 3 r (3 2 − 3 − 1) + c 2 φ r ( φ 2 − φ − 1) φ 2 − ¯ φ − 1) + c 4 (1 2 − 1 − 1) + c 3 ¯ φ r (¯ = c 1 3 r · 5 − c 4 We can plug in r = 0 and r = 1 and compute sums to solve and find that c 1 = 2 5 . Hence s { 0 , 2 , 3 } ( r ) s A ( r ) = 2 lim |A| r = lim 5 . 4 r r →∞ r →∞ Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Proof Let g ( λ ) := det( M − λ I ) be the characteristic polynomial of M with eigenvalues λ 1 , λ 2 , . . . , λ y , where each λ i has multiplicity e i , so a z +1 � α k λ k . g ( λ ) = k =0 By Cayley-Hamilton, we know that g ( M ) = 0. Thus we have a z +1 � α k M k 0 = g ( M ) = k =0 and hence, for all r , � a z +1 a z +1 � � � α k M k 0 = ω r ( m ) = α k ω r + k ( m ) . k =0 k =0 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Let I r = { 2 r , 2 r + 1 , 2 r + 2 , . . . , 2 r +1 − 1 } . Then I r = 2 I r − 1 ∪ (2 I r − 1 + 1). Thus 2 r +1 − 1 � s A ( r ) = f A ( n ) n =2 r 2 r − 1 � = f A (2 n ) + f A (2 n + 1) n =2 r − 1 2 r − 1 � = f A ( n ) + f A ( n − b 2 ) + · · · + f A ( n − b s ) n =2 r − 1 + f A ( n − c 1 ) + · · · + f A ( n − c t ) . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Now 2 r − 1 2 r − 1 k f A (2 r − 1 − j ) − f (2 r − j ) � � � � � f A ( n − k ) = f A ( n ) + , n =2 r − 1 n =2 r − 1 j =1 so 2 r − 1 � s A ( r ) = |A| f A ( n ) + h ( r ) = |A| s A ( r − 1) + h ( r ) , n =2 r − 1 where h r is such that a z +1 � α k h ( r + k ) = 0 . k =0 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
The solution to this inhomogeneous recurrence relation is of the form y s A ( r ) = c 1 |A| r + � p i ( λ i ) , i =1 where p i ( λ i ) = � e i j =1 c ij r j − 1 λ r i . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
We can compute � a z +1 k =0 α k s A ( r + k ), and for sufficiently large r , we have a z +1 a z +1 α k |A| r + k + 0 = c 1 |A| r g ( |A| ) . � � α k s A ( r + k ) = c 1 k =0 k =0 Then we can solve for c 1 to see that � a z +1 k =0 α k s A ( r + k ) c 1 = . |A| r g ( |A| ) Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
A c ( A ) N( c ( A )) A c ( A ) N( c ( A )) 4 { 0 , 1 , 2 } 1 1 . 000 { 0 , 1 , 3 } 0 . 800 5 5 14 { 0 , 1 , 4 } 0 . 625 { 0 , 1 , 5 } 0 . 560 8 25 425 176 { 0 , 1 , 6 } 0 . 499 { 0 , 1 , 7 } 0 . 450 852 391 137 1448 { 0 , 1 , 8 } 0 . 405 { 0 , 1 , 9 } 0 . 384 338 3775 1990 3223 { 0 , 1 , 10 } 0 . 360 { 0 , 1 , 11 } 0 . 340 5527 9476 2020 47228 { 0 , 1 , 12 } 0 . 322 { 0 , 1 , 13 } 0 . 306 6283 154123 35624 699224 { 0 , 1 , 14 } 0 . 291 { 0 , 1 , 15 } 0 . 280 122411 2501653 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
˜ c ( ˜ A c ( A ) A A ) 7 3 { 0 , 1 , 2 , 4 } { 0 , 2 , 3 , 4 } 11 11 2531 1344 { 0 , 2 , 3 , 6 } { 0 , 3 , 4 , 6 } 9536 9536 3401207 1156032 { 0 , 1 , 6 , 9 } { 0 , 3 , 8 , 9 } 16513920 16513920 132416 51145 { 0 , 1 , 7 , 9 } { 0 , 2 , 8 , 9 } 655040 655040 4044 6716 { 0 , 4 , 5 , 6 , 9 } { 0 , 3 , 4 , 5 , 9 } 83753 83753 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Theorem Let A , f A ( n ) and M = [ m α,β ] be as above. Define ˜ A := { 0 , a z − a z − 1 , . . . , a z − a 1 , a z } . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Theorem Let A , f A ( n ) and M = [ m α,β ] be as above. Define ˜ A := { 0 , a z − a z − 1 , . . . , a z − a 1 , a z } . Let N = [ n α,β ] be the ( a z + 1) × ( a z + 1) matrix such that f ˜ A (2 n ) f ˜ A ( n ) A (2 n − 1) A ( n − 1) f ˜ f ˜ = N . . . . . . . A (2 n − a z ) A ( n − a z ) f ˜ f ˜ Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Theorem Let A , f A ( n ) and M = [ m α,β ] be as above. Define ˜ A := { 0 , a z − a z − 1 , . . . , a z − a 1 , a z } . Let N = [ n α,β ] be the ( a z + 1) × ( a z + 1) matrix such that f ˜ A (2 n ) f ˜ A ( n ) A (2 n − 1) A ( n − 1) f ˜ f ˜ = N . . . . . . . A (2 n − a z ) A ( n − a z ) f ˜ f ˜ Then m α,β = n a z − α, a z − β . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Proof Recall we can write A := { 0 , 2 b 1 , . . . , 2 b s , 2 c 1 + 1 , . . . , 2 c t + 1 } , so that f A (2 n − 2 j ) = f A ( n − j ) + f A ( n − j − b 1 ) + · · · + f A ( n − j − b s ) and f A (2 n − 2 j − 1) = f A ( n − j − c 1 − 1) + · · · + f A ( n − j − c t − 1) for j sufficiently large. Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Then m α,β = 1 ⇐ ⇒ f A ( n − β ) is a summand in the recursive sum that expresses f A (2 n − α ) ⇐ ⇒ 2 n − α = 2( n − β ) + K , where K ∈ A ⇐ ⇒ 2 β − α ∈ A . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Now n a z − α, a z − β = 1 ⇐ ⇒ f ˜ A ( n − ( a z − β )) is a summand in the recursive sum that expresses f ˜ A (2 n − ( a z − α )) ⇒ 2 n − ( a z − α ) = 2( n − ( a z − β )) + ˜ K , where ˜ K ∈ ˜ ⇐ A ⇒ a z + α − 2 β = ˜ ⇐ K ⇐ ⇒ 2 β − α ∈ A . Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Thus M = A − 1 NA , where 0 0 · · · 0 1 0 0 · · · 1 0 . . . . . . . . A = , . . . . 0 1 · · · 0 0 1 0 · · · 0 0 so M and N have the same characteristic polynomial. Hence the denominator of c ( A ) is the same as the denominator of c ( ˜ A ). Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
Future Research Ideas ◮ Finding more families of robust polynomials ◮ Determining the cluster points of � � β 1 ( f ) β 0 ( f ) + β 1 ( f ) : f ( x ) is a polynomial ◮ Finding formulas for c ( A ) ◮ Exploring properties of f A ( n ) in bases other than 2 Katie Anders UTTyler Odd behavior in the coefficients of reciprocals of binary power series
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