Roots of Discrete Analytic Polynomials Susan Durand, Caitlin Still - PowerPoint PPT Presentation

Roots of Discrete Analytic Polynomials Susan Durand, Caitlin Still Mentor - Dr. Dan Volok SUMaR at K-State July 21, 2015 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 1 / 26

Introduction The Lattice We are working on the integer lattice in the plane Λ = { ( x , y ) : x , y ∈ Z } ⊂ R 2 = { x + iy : x , y ∈ Z } ⊂ C Two vertices on Λ are considered to be adjacent if | ( x 1 + iy 1 ) − ( x 2 + iy 2 ) | = 1. Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 2 / 26

Introduction DA Functions A function f : Λ → C is considered to be Discrete Analytic if it satisfies the following Cauchy-Riemann equation: f ( z + 1 + i ) − f ( z ) = f ( z + i ) − f ( z + 1) . (1 + i ) ( i − 1) f ( z + 1 + i ) z + i z + 1 + i f ( z + i ) f z f ( z ) z + 1 f ( z + 1) Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 3 / 26

Introduction Integration Definition of Integration Integration on Λ is defined as follows m � f ( z n ) + f ( z n − 1 ) � f δ z = ( z n − z n − 1 ) 2 γ n =1 for γ ∈ Λ, where γ = ( z 0 , z 1 , ..., z n ). � A function f is considered DA if and only if γ f δ z = 0 for every loop γ . Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 4 / 26

Introduction Discrete Analytic Polynomials Definition of Polynomials A function f : Λ − → C is a polynomial if it can be written as n a j , k x j y k , with a j , k ∈ C f = � j , k =0 We are working with polynomials in x and y with complex coefficients, not just z . 1 , z , and z 2 are DA, but z 3 is not DA. Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 5 / 26

Introduction Problem Find an estimate of the number of roots of DAPs of degree d. � 1+ i � 2 i − 1 � i z 3 − z 2 + z − iy − y � � � Example: P ( z ) = 6 2 6 6 p (0) = p (1) = p ( i ) = p (1 + i ) = 0 , p (2) = 1 1 + i i 0 1 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 6 / 26

Finding a Bound Theorem Let p 0 ( x ) be a polynomial in x, then there exists a unique DAP p ( x , y ) such that p ( x , 0) = p 0 ( x ) . Definition of a Polynomial Basis by Extension z n ( x , y ) is determined by z n ( x , 0) = x n Therefore, a DAP can be written in the following way: d � p ( x , y ) = a k z k ( x , y ) k =0 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 7 / 26

Finding a Bound ( p ⊙ q )( x , y ) is determined by ( p ⊙ q )( x , 0) = p ( x , 0) q ( x , 0) z 0 = 1 z 1 = z z 2 = z 2 . . . z ⊙ z n = z n +1 = ( z ⊙ ) n z � � f ( x , y +1)+ f ( x , y − 1) z ⊙ f = xf ( x , y ) + iy 2 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 8 / 26

Finding a Bound Finitely Many Roots z n ( x , y ) = z n + lower order terms A DAP of degree d is dominated by c · z d , so a limit argument shows that a DAP has finitely many roots because there are no roots outside of some sufficiently large circle. Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 9 / 26

If p is a DAP such that p ( z ) = 0 , z ∈ R , where R is an a × b rectangle, then p = 0 or deg ( p ) > a + b If S = { s 0 , s 0 + 1 , ... s 0 + a , s 0 + i , ... s 0 + bi } and p ( z ) = 0 on S , then p = 0 or deg ( p ) ≥ # S s 0 + ib s 0 s 0 + a Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 10 / 26

Finding the Bound Uniqueness Sets  P ( s 0 ) = c 0     z 0 ( s 0 ) z 1 ( s 0 ) · · · z d ( s 0 )      a 0 c 0    .  . a 1 c 1  .   z 0 ( s 1 ) .      .   . ⇐ ⇒  = . .     . .   . . . .     . .  . .          a d c d  z 0 ( s d ) · · · · · · z d ( s d )     P ( s d ) = c d  Definition of Uniqueness Set � d � A set S = { s 0 , ..., s d } is a uniqueness set if z j ( s i ) i , j =0 is invertible. Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 11 / 26

Theorem Let S = { s 0 , ... s d } ⊂ Λ then the following are equivalent: 1 S is a uniqueness set. 2 If p ( z ) = 0 for every point in S , then p = 0 or deg ( p ) > d . 3 For all ( c 0 , c 1 , ..., c d ) ∈ C d +1 , there exists a unique p ( z ), such that p ( s k ) = c k and either p = 0 or deg ( p ) ≤ d . Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 12 / 26

Finding the Bound Uniqueness Sets Observation Let R be an a × b rectangle, and let S ⊂ R be a uniqueness set. Then # S ≤ a + b + 1. Observation If Q is the set of all zeros for a DAP p ( z ), and S ⊂ Q is a uniqueness set, then # S ≤ d . Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 13 / 26

Finding the Bound Understanding Uniqueness Sets Figure: Not Uniqueness Sets Figure: Uniqueness Sets Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 14 / 26

Finding the Bound Uniqueness Set Theorem Theorem Let S = { s 0 , ... s d } ⊂ Λ then the following are equivalent: 1 S is a uniqueness set. 2 For all ( c 0 , c 1 , ... c d ) ∈ C d +1 there exists a DA f such that f ( s k ) = c k . 3 For all a × b rectangles R , #( S ∩ R ) ≤ a + b + 1. Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 15 / 26

Finding the Bound Q is the set of all roots of a DAP p of degree d. S is the maximal uniqueness subset of Q . Choose disjoint rectangles, R k , so that #( R k ∩ S ) = a k + b k + 1. n Q ⊂ � R k k =1 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 16 / 26

Finding the Bound Q is the set of all roots of a DAP p of degree d. S is the maximal uniqueness subset of Q . Choose disjoint rectangles, R k , so that #( R k ∩ S ) = a k + b k + 1. n Q ⊂ � R k k =1 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 16 / 26

Finding the Bound Q is the set of all roots of a DAP p of degree d. S is the maximal uniqueness subset of Q . Choose disjoint rectangles, R k , so that #( R k ∩ S ) = a k + b k + 1. n � Q ⊂ R k k =1 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 16 / 26

Finding the Bound Q is the set of all roots of a DAP p of degree d. S is the maximal uniqueness subset of Q . Choose disjoint rectangles, R k , so that #( R k ∩ S ) = a k + b k + 1. n Q ⊂ � R k k =1 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 16 / 26

Finding the Bound Result n � # S = ( a k + b k + 1) ≤ d k =1 n � # Q ≤ ( a k + 1)( b k + 1) k =1 Theorem If p is a DAP of deg ( p ) = d > 0 and Q is the set of the zeros of p , then # Q ≤ ( d +1 2 ) 2 . Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 17 / 26

Future Research Rhombic Lattice Figure: G = ( V , E ) Figure: Constructing the Dual Graph Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 18 / 26

Future Research Rhombic Lattice Figure: G = ( V , E ) Figure: Constructing the Dual Graph Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 18 / 26

Future Research Rhombic Lattice Figure: G = ( V , E ) Figure: G ∗ = ( V ∗ , E ∗ ) Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 18 / 26

Figure: Combined Graph G ⋄ = ( V ∪ V ∗ , E ⋄ ) Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 19 / 26

Future Research Weighted Discrete Analytic Definition of Discrete Analytic A function f : V ∪ V ∗ → C is discrete analytic if it satisfies f ( h e ) − f ( t e ) = i f ( r e ) − f ( l e ) W e W e ∗ where W e ∗ and W e are positive weights. h e l e e e ∗ r e t e Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 20 / 26

Future Research Integration For integration, we require a DA function z : V ∪ V ∗ → C . Definition of Integration n f ( t k )+ f ( t k − 1 ) � � For a path γ on the graph, γ f δ z = ( z ( t k ) − z ( t k − 1 )) 2 k =1 Then f : V ∪ V ∗ → C is DA if and only if � γ f δ z = 0 for every loop γ . Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 21 / 26

Future Research Multiplication We arbitrarily chose a point so that 0 ∈ V ∪ V ∗ . We set � z ( z ⊡ f )( z ) = f (0) − f ( z ) + f δ z . 2 0 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 22 / 26

Future Research In order for this multiplication operator to preserve discrete analyticity, z has to satisfy: z ( t e ) + z ( h e ) = z ( r e ) + z ( l e ) Thus, z ( G ⋄ ) is a rhombic lattice. Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 23 / 26