Re: Number of Roots of Polynomials in R[x], R Has Zero Divisors

quasi <quasi@xxxxxxxx> writes:

On Wed, 25 Apr 2007 00:12:33 -0500, quasi <quasi@xxxxxxxx> wrote:

On Tue, 24 Apr 2007 15:41:19 EDT, Jonathan Groves <JGroves@xxxxxxxxxx>
wrote:

Dear All,

It is well-known that if p(x) is a polynomial of degree d in R[x] where R
is an integral domain, then p(x) has at most d roots (in fact, exactly d
roots if you count a root of multiplicity m as m roots). In fact, this
is still true if R is noncommutative, as long as R has no zero divisors.


However, when we remove the no zero divisors assumption, then this result
is no longer true. For example, let R=Z/(6), and consider the polynomial
p(x)=x^2+x. Then p(x) has roots 0, 2, 3, and 5 (mod 6). Thus, p(x) has
degree 2 yet has four distinct roots.

Does anyone know how to tell how many roots an arbitrary p(x) has when R
has zero divisors? Of course, if R is small, then it is not too much
trouble to evaluate p(x) for all elements in R, but this is tedious if R
is large, such as Z/(500), and cannot be done if R is infinite. Has such
a theorem been found? I guess so, but I have never seen it before.

Thanks for your answers.

Jonathan Groves

It's possible to have a polynomial with infinitely many zeros.

Let R be the ring of n x n matrices over an infinite field and let D
in R be a nonzero diagonal matrix.

Let T be the set of all lower triangular matrices whose diagonal is
the same as D.

Let p(x) be the characteristic polynomial of D. Then p(A)=0 for all A
in T, hence p has infinitely many zeros in R.

Two followup questions:

(1) If R is a commutative ring with 1, does every nonzero polynomial
p in R[x] have at most finitely many zeros in R?

No. Let S be some infinite set and K any commutative ring with cardinality
at least 2, and consider the ring R = K^S of functions from S to K, with
operations defined pointwise. We identify K with the subring of R
consisting of constant functions. If a and b are distinct members of K,
the polynomial (x-a)(x-b) has infinitely many zeros in R, namely those
functions whose values are all in {a,b}.

(2) Does there exist an infinite ring R with 1, not necessarily
commutative, and a polynomial p in R[x] such that all elements of R
are zeros of p?

In problem (2), above, I meant, of course, a nonzero poynomial p.

In the example above, let K be finite and
p(x) = product_{k in K} (x-k).
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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