Exact degree of a symmetric Boolean function Rajat

Exact degree of a symmetric Boolean function Rajat Mittal (IIT Kanpur)

Boolean functions •

Polynomial representation Multilinear (degree one in each variable) Unique •

Degree of a representation

Symmetric function •

non-constant What is the minimum possible degree of a Boolean symmetric function?

Same Degree

Equivalence: multivariate and univariate

Outline Problem description Known results Best bound (GR 97) Random function (GR 97) More than two values (CST 10) Conjecture Number theory approaches Possible approaches Dual witness

What is the minimum possible degree of a non-constant Boolean symmetric function?

Finding the coefficients •

Matrix equation |x|=j

Taking inverse j

Distance between primes

More than two outputs

Upper bound 2 Lower bound 3 Trivial case 0 0 1

Outline Problem description Known results Best bound (GR 97) Random function (GR 97) More than two values (CST 10) Conjecture Number theory approaches Possible approaches Dual witness

Results

Conjecture

Experimental results (GR 97) •

Outline Problem description Known results Best bound (GR 97) Random function (GR 97) More than two values (CST 10) Conjecture Number theory approaches Possible approaches Dual witness

Good sequences

Good sequences

Conjecture for school students At what level does this become a constant sequence?

Outline Problem description Known results Best bound (GR 97) Random function (GR 97) More than two values (CST 10) Conjecture Number theory approaches Possible approaches Dual witness

Approximate degree Primal Dual

Dual Witness •

Witness: more formulations •

Almost Vandermonde identity j i

Conclusion •

THANKS