# Jason Polak, PhD I am a mathematician. You can find my research papers here, as well as my blogs and some random other things.

What kind of research do I do? Here's an idea: did you know that the polynomial function f(x) = x3 + 6x2 + x permutes the elements of Z/9 (the integers modulo nine)? Not all polynomials permute the elements of the underlying coefficient ring. The ones that do are called permutation polynomials, and for finite coefficient rings the corresponding permutations form a group! Check out this recent paper to learn more about these polynomials. I also like primes. Do you want to check whether a number is prime? Use this Miller-Rabin utility I wrote:

Other than that, I'm interested in algebra, finite rings and combinatorics, and related topics. I like to focus on concrete problems, often with a computational angle. Speaking of computational things: are you interested in finding colourful patterns? Try the game Znocks, my Znax clone written in Javascript.

## Papers

6The Ring Structure on the Grothendieck Group of Commuting Endomorphisms over a Field
For a given ring k, the category of k[t]-modules that are finitely generated and projective as k-modules form an exact category. We compute the natural ring structure on the Grothendieck group of this category when k is a perfect field, extending a result of Almkvist for algebraically closed fields.
Submitted 2017-08-31!
5The Polypermutation Group of an Associative Ring
We define a new invariant called the polypermutation group of a finite ring, which is the group of permuations that can be written as polynomials over the given ring. We derive some basic properties of this group and compute it for Z/p2 where p is a prime number.
Submitted 2017-07-02!
4Counting Separable Polynomials in Z/n[x]
Using the theory of separable algebras, we derive an easy to use formula for the number separable polynomials in Z/N[x] of given degree.
Published online in the Canadian Mathematical Bulletin 2017-04-13
3A Cotriple Construction of a Simplicial Algebra Used in the Definition of Higher Chow Groups
Bloch used a simplicial complex to define higher Chow groups. We define a cotriple whose associated simplicial complex is none other than Bloch's.
Published in Theory and Applications of Categories, Vol. 31, No. 13 (2016), pp. 384-387
2Exposing Relative Endoscopy in Unitary Symmetric Spaces
Relative endoscopy is a conjectural phenomenon that may help in the understanding of relative trace formula, which in turn is a fundamental tool for studying distinguished automorphic representations. This paper provides evidence for the existence of relative endoscopy via a concrete p-adic calculation.
Published in Res. Math. Sci. Vol. 2. No. 1
1A Note on VH Subdivisions (with Daniel T. Wise)
A group is residually finite if every nontrivial element survives in some finite quotient. In this paper we give a geometric 2-complex tiling method to determine whether a given group is residually finite. We then use our method to answer a question of Ian Leary.
Published in Publ. Mat. 57 (2013), pp. 421-428.

## Contact

You can contact me at . Consider using my PGP public key to encrypt the mail (see how to do this).

Key fingerprint: F50B 455E F406 CA05 DBED 673F 6C62 3CB6 1BA5 D66C The Yellow-tailed Black Cockatoo (Zanda funereus). Photo taken by Jason in Yarra Bend Park, Melbourne.