Unreachable functions - Missouri University of Science and
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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.