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} ⊂ R2 = {x + iy : x, y ∈ Z} ⊂ C Two vertices on Λ are considered to be adjacent if |(x1 + iy1) − (x2 + iy2)| = 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) (1 + i) = f (z + i) − f (z + 1) (i − 1) .

z + i z + 1 + i z z + 1 f f(z + i) f(z + 1 + i) f(z) 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

• γ

f δz =

m

• n=1

f (zn) + f (zn−1) 2 (zn − zn−1) for γ ∈ Λ, where γ = (z0, z1, ..., zn). A function f is considered DA if and only if

• γ f δz = 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 f =

n

• j,k=0

aj,kxjyk, with aj,k ∈ C We are working with polynomials in x and y with complex coefficients, not just z.

1, z, and z2 are DA, but z3 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. Example: P(z) = 1+i

6

• z3 −

i

2

• z2 +

2i−1

6

• z − iy−y

6

p(0) = p(1) = p(i) = p(1 + i) = 0, p(2) = 1

i 1 + i 1 Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 6 / 26

Finding a Bound

Theorem Let p0(x) be a polynomial in x, then there exists a unique DAP p(x, y) such that p(x, 0) = p0(x). Definition of a Polynomial Basis by Extension zn(x, y) is determined by zn(x, 0) = xn Therefore, a DAP can be written in the following way: p(x, y) =

d

• k=0

akzk(x, y)

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) z0 = 1 z1 = z z2 = z2 . . . z ⊙ zn = zn+1 = (z⊙)nz z ⊙ f = xf (x, y) + iy

• f (x,y+1)+f (x,y−1)

2

• Susan Durand, Caitlin Still (SUMaR)

Roots of Discrete Analytic Polynomials July 21, 2015 8 / 26

Finding a Bound

Finitely Many Roots

zn(x, y) = zn+ lower order terms A DAP of degree d is dominated by c · zd, 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 = {s0, s0 + 1, ...s0 + a, s0 + i, ...s0 + bi} and p(z) = 0 on S, then p = 0 or deg(p) ≥ #S

s0 s0 + a s0 + ib

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 10 / 26

Finding the Bound

Uniqueness Sets

                   P(s0) = c0 . . . ⇐ ⇒       z0(s0) z1(s0) · · · zd(s0) z0(s1) . . . . . . . . . z0(sd) · · · · · · zd(sd)            a0 a1 . . . ad      =      c0 c1 . . . cd      P(sd) = cd Definition of Uniqueness Set A set S = {s0, ..., sd} is a uniqueness set if

• zj(si)

d

i,j=0 is invertible.

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 11 / 26

Theorem Let S = {s0, ...sd} ⊂ 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 (c0, c1, ..., cd) ∈ Cd+1, there exists a unique p(z), such that

p(sk) = ck 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 = {s0, ...sd} ⊂ Λ then the following are equivalent:

1 S is a uniqueness set. 2 For all (c0, c1, ...cd) ∈ Cd+1 there exists a DA f such that f (sk) = ck. 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, Rk, so that #(Rk ∩ S) = ak + bk + 1. Q ⊂

n

• k=1

Rk

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, Rk, so that #(Rk ∩ S) = ak + bk + 1. Q ⊂

n

• k=1

Rk

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, Rk, so that #(Rk ∩ S) = ak + bk + 1. Q ⊂

n

• k=1

Rk

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, Rk, so that #(Rk ∩ S) = ak + bk + 1. Q ⊂

n

• k=1

Rk

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 16 / 26

Finding the Bound

Result

#S =

n

• k=1

(ak + bk + 1) ≤ d #Q ≤

n

• k=1

(ak + 1)(bk + 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 (he) − f (te) We = i f (re) − f (le) We∗ where We∗ and We are positive weights.

he te le re

e 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 For a path γ on the graph,

• γ f δz =

n

• k=1

f (tk)+f (tk−1) 2

(z(tk) − z(tk−1)) Then f : V ∪ V ∗ → C is DA if and only if

• γ f δz = 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 ⊡ f )(z) = f (0) − f (z) 2 + z f δz.

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(te) + z(he) = z(re) + z(le) Thus, z(G ⋄) is a rhombic lattice.

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 23 / 26

Future Research

A Polynomial Basis on G ⋄

Polynomial Basis on G ⋄ Set ζ0 = 1, ζn+1 = z ⊡ ζn In case of Λ, ζn are determined by ζn(x, 0) = x

n

• , where

x n

• = x(x − 1)...(x − n + 1)

n!

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 24 / 26

Future Research

p : V ∪ V ∗ ⇒ C is a polynomial if p =

d

• k=0

akζk, where ak is a complex number. Problem Determine a bound for the number of roots of a DAP of degree d.

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 25 / 26

References:

• R. J. Duffin.

Basic properties of discrete analytic function. Duke Math. J., 23, 335-363 (1956).

• D. Alpay, P. Jorgensen, R. Seager, and D. Volok.

On discrete anayltic functions: Products, rational functions and producing kernels. JAMC, 41, 393-426 (2013).

• L. Lov´

asz. Discrete analytic functions: An exposition. International Press, 241-273 (2004).

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 26 / 26

Acknowledgments

This research was done with funding from NSF under DMS award number 1262877. We would like to thank Dr. Korten and Dr. Yetter for organizing K-State’s SUMaR program. We would especially like to thank Dr. Dan Volok for being our mentor.

Susan Durand, Caitlin Still (SUMaR) Roots of Discrete Analytic Polynomials July 21, 2015 26 / 26