Polynomially Inexpressible Functions on Rings
by Matt Insall
University of Missouri – Rolla
Let R be a ring, and let f be a function from R to R We say that f is polynomially
expressible provided that there is a permutation g on the underlying set of R such that the
composite function h = g-1ofog is a polynomial function on the ring R.
More generally, if A is an algebraic system, then a function f on A is polynomially
expressible provided that there is a bijection g from A to A such that the conjugate of A
by g is a polynomial function on the algebra A. The problem of determining which
functions on an algebra are polynomially expressible is motivated by the problem of
determining which problems can be solved by a computation device, in the following
way: The operations of the algebra are the fundamental operations in the computing
device. The elements of the algebra are the values with which the computing device can
compute. The bijections are the functions that determine an assignment of the values to
storage locations, in a one-to-one correlation. The author has worked on this type of
problem since the early nineteen-nineties (see [I], [IW] and [MIW]). In the case of rings,
the problem of generating examples with which to work appears to be computationally
intractable in the finite case, and reduces to questions about foundational limitations in
the infinite case (even for fields). In the finite case, however, it is known that each finite
field is polynomially complete, so that for n a prime, every function on any n-element set
is expressible as a polynomial over Zn, and for n a prime power, every function on an nelement set is expressible as a polynomial over the n-element field. In the case n=4, the
function that maps 0 to 3, 1 to 3, 2 to 2 and 3 to 1 is not polynomially expressible over
the ring Z4. (It is, however, expressible as a polynomial over the four-element field, of
course.) We are currently working on the problem of which functions are polynomially
expressible over the ring Z6, in hopes that after enough computational small examples,
we will see a pattern that will be of use in the cases of Zn for arbitrary composite n. (We
have proved, however, that for any finite ring that is not a field, there is a function on the
ring that is not a polynomial function. The problem of proving that some function is not
polynomially expressible is a harder problem.)
Recall that for each positive natural number n, the nth roots of unity in the complex plane
form a group under multiplication, and this group is isomorphic to the additive group Zn,
due to an appropriate action of the integers on the set of nth roots of unity. But in fact, the
ring Zn of integers modulo n acts on this group, in the exponents, in such a way that any
operation on the set of nth roots of unity that can be achieved by multiplication is
obtainable by exponentiation: a polynomial with integer coefficients in the exponent on a
generator of the group will suffice, and in fact, the polynomial may be taken to be a
polynomial over Zn. Thus, when n is composite, there is a function on the set of nth roots
of unity that cannot be so expressed without first permuting the roots of unity and then
reversing the permutation after applying the polynomial exponent.
We intend to show, among other things, that for every ring that is not polynomially
complete, there is a function on the ring that is not polynomially expressible. (The
following related situation supports, but does not prove, our conjecture: Given a finite
algebra A, if every function, of every arity, on A, is a conjugate of a polynomial on A,
then A is polynomially complete. This result is discussed in [IW]. It was proved by
Kearnes in a private communication.) We intend to describe ways of generating such
polynomnially inexpressible functions, and we plan to describe how to obtain such
functions in more general algebraic systems, including dioids, which are the subject of
study in another effort of the author, in conjunction with a colleague in Mechanical
Engineering. Currently, a graduate student in the Mathematics and Statistics Department
is working with the author on some of the computational and theoretical issues related to
the Zn case, for small n.
The author’s goal, with respect to this funding request, is to obtain salary support in the
summers for several years, travel funds to attend conference on related topics, and
support for graduate students in mathematics and computer science to work on these
problems for rings. After enough of the details for rings are known, we will expand the
results to cover, first, those algebras that are polynomially equivalent to rings, and then
we will expand the investigation further into the territory occupied by those algebras that
are not polynomially equivalent to rings.
References
[I] M. Insall, ``By Renaming if Necessary’’. Under Review.
[IW] M. Insall and Ralph Wilkerson, ``Polynomial Completeness and Conjugation’’.
Under Review.
[MIW] L. Mullin, M. Insall and R. Wilkerson, ``Conjugating Polynomials on Finite
Rings'', in the 1994 ACM SigApp Symposium on Applied Computing.