1. Anne Calder & Alan Cordero

2. Goals Create 1000 random polynomials up to degree 100 Calculate the number of real zeros on average for each degree Check our results with the Kac Formula Change the way our coefficients are chosen and check the average number of real zeros Create random matrices and check the number of real eigenvalues

3. Review Kac Formula for finding the expected amount of real zeros of a random polynomial Coefficients are chosen from a standard normal distribution

4. Random Polynomial Example Create a random polynomial of degree 5 >> x=randn([1 6]) -1.2025 0.4054 0.1872 1.1233 -1.3993 0.2030 What polynomial would look like -1.2025x 5 +0.4054x 4 +0.1872x 3 +1.1233x 2 -1.3993x+0.2030=0

5. Roots of the polynomial >> roots(x) ans = -0.7096 + 0.8911i -0.7096 - 0.8911i 0.7938 + 0.3759i 0.7938 - 0.3759i 0.1687

6. Simulating 1000 random polynomials N=1000; % 1000 simulations x=zeros(1000, 5); % creates a empty 1000 x 5 matrix for k=1:N x(k,:) = roots(randn([1 6])); % calculates the roots end after each simulation size(x(imag(x)==0)) /1000 % displays the average number of real zeros

7. Kac Plot Comparison N=100; En = zeros(1,N); for i=1:N En(i) = 4/pi*quad(@(x)sqrt(1./(1- x.^2).^2 - (i+1)^2*x.^(2*i)./(1- x.^(2*i+2)).^2),0,1); end plot(1:N,En)

8. Kac Plot Comparison

9. Proportion of Real Zeros Comparison

11. Different Coefficients rand – a random number from the standard uniform distribution on the open interval (0,1) rand -.5 – a random number from the uniform distribution on the open interval (-.5,.5)

12. Graphs of Different Coefficients

13. How many eigenvalues of a random matrix are real? The expected number of real eigenvalues with independent standard normal entries is:

15. Eigenvalue Code Simulating 1000 Random Matrices n=1000; Y=zeros(1,n); %creates an empty 1:n empty matrix X=[1:n]; for i=1:n A=randn(i); %creates a matrix size n with random entries e=eig(A); %eigenvalues of a s=size(e(imag(e)==0)); %amount of real eigenvalues p=s(1) Y(i)=p; plot(X,Y,'b.') title('Amount of real eigenvalues') xlabel('n - Size of square matrix') ylabel('Amount of real eigenvalues') end

16. The data doesn’t change dramatically with the change of how the coefficients are chosen.

18. Proportion of Real Eigenvalues Comparison

19. Questions?