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I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)

Preamble

We consider polynomials f ∈ ℤ[x] with roots in ℝ, and for each polynomial f, the principal root is the real root with the largest magnitude. In the case of two roots of equal magnitude, we take the positive one. So, for instance, √5 is the principal root of x2 − 5, the golden ratio φ is the principal root of x2 − x − 1, and −φ is the principal root of x2 + x − 1.

 Edit: Previously, I had defined the principal root to be the maximal one; I revised this definition based on remarks by Kevin Buzzard below. This also motivates some revisions to the questions I ask. ]

It's tempting to think that we could represent algebraic numbers by (minimal) polynomials over ℤ for which they are the principal root. We implicitly do this with rational numbers all the time: a/b denotes the real number which is the principal root of bx − a, and can formally be defined in such terms. This is also precisely what we do with algebraic integers: nb is defined to be the principal root of xn − b, at least for b non-negative. This approach seems problematic for negative algebraic numbers such as φ−1, whose minimal polynomial is x2 + x − 1, which is the same as for −φ; thus every polynomial for which φ−1 is a root will also have the (larger in magnitude) root −φ. A similar problem arises for −√5, of course. But let us focus on algebraic numbers which are the principal roots of their minimal polynomials.

Questions

For two irredicible polynomials f,g ∈ ℤ[x] with roots in ℝ, let u,v ∈ ℝ be their principal roots.

  1. Are there broad classes of polynomials f and g, including ones of degree 2 or more, of course, for which there is a "simple" formula (e.g. involving no recursive functions more complicated than sums, products, exponents, and "well-known" number sequences) for the minimal polynomials of u + v, uv, or uv−1?
  2. Does there exist such a "simple" formula for some polynomial (not necessarily irredicuble) for which u + v, uv, or uv−1 is the principal root?
  3. If such questions are a proper subject of some body of research or well-studied theory: what is the name of the associated field of mathematics? (E.g. is this a special topic of Galois theory?)

Note that it is unlikely that we can obtain any sort of satisfactory answer for obtaining a minimal polynomial for u − v ; for instance, if u = √2 and v = √3, then the minimal polynomial of all four numbers ± √3 ± √2 have the same minimal polynomial, x4 − 26x2 + 145. For similar reasons, it is unlikely that there is a complete solution for arbitrary sums of principal roots: if u is a principal root of f(x), then −u is a principal root of f(−x), which differs from f in the case that f is not an even function of x. Thus, some differences of principal roots may also be expressed as sums of principal roots.

The difficulties described above and in the preamble suggest that a clean and elegant theory is unlikely; but I'm hoping that there are interesting classes of algebraic numbers which may be treated in this way.

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2  
Regarding Q2: Are you aware of the notion of the conjugates of an algebraic number? If alpha is algebraic then any polynomial f in Z[x] with alpha as a root will have all the conjugates of alpha as roots as well. Hence your 2nd question seems to be based on a misapprehension. If u has a conjugate u' with u'<0<u and if v has a conjugate v' with v'<0<v then u'v' will be a conjugate of uv and it might be bigger than uv, and in this case uv can never be the principal root of any polynomial at all. An explicit example would be u=sqrt(2)-1 and v=sqrt(3)-1. –  Kevin Buzzard Apr 15 '10 at 8:29
    
I am aware of this concept; indeed I am implicitly invoking it for the relationship between the negation and the inverse of the golden ratio, as well as my comments about differences of principal roots after my questions. You make a good point about products of "large", negative, non-principal roots. (I'm not sure how it relates to question 2 in particular, as opposed to question 1.) I will make a revision to the question, in light of this problem –  Niel de Beaudrap Apr 15 '10 at 9:06
    
I think I understand why you were addressing Q2 as opposed to Q1, and I can answer. While there exist reducible polynomials whose principal roots are u+v, uv, etc. only if there exist irreducible ones, I allowed for the possibility that sometimes the reducible ones might be easier to find than the irreducible ones. (That is, lifting the requirement that the polynomial be irreducible can only make the problem of constructing a suitable polynomial easier.) –  Niel de Beaudrap Apr 15 '10 at 11:56

3 Answers 3

up vote 2 down vote accepted

If $u$ is a root of $f$ and $v$ is a root of $g$ then $u+v$ is a root of the resultant of $f(x-y)$ and $g(y)$ (for the purpose of calculating the resultant, we take these as polynomials in $y$). The resultant is just a determinant of a matrix whose entries are all coefficients of the polynomials, so it would seem to satisfy your request for a simple formula.

There are similar formulas for polynomials with $uv$ as a root, or $uv^{-1}$.

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All of this has been fully worked out by Marc Rybowicz, bits of which are available in a JSC paper (preprint available as On the normalization of numbers and functions defined by radicals. This is all implemented in Maple's evala routine for normalization of algebraic numbers, and are known as indexed RootOfs. The roots of index 1 are exactly those you define above.

Unfortunately, I don't think that Marc has published everything he knows about this topic. You might wish to contact him if you really want to know all the details.

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I will have to review this aritcle, as the terminology seems likely to be pertinent to my question, but without it being obvious (to me) that it answers it. Thanks for the reference! –  Niel de Beaudrap Apr 15 '10 at 15:05

I know of one class of polynomials for which there is something like an answer to your question. It is the class of polynomials which are characteristic polynomials of symmetric matrices with non-negative integer coefficients. These matrices describe adjacency matrices of undirected graphs, and there are two well-known constructions which take two such graphs $G, H$ whose characteristic polynomials have "principal roots" $u, v$ and returns graphs whose characteristic polynomial has principal roots $uv, u + v$. The first construction is the tensor product of graphs, and the second is the Cartesian product. In matrix terms these correspond to the Kronecker product and Kronecker sum of matrices.

Some keywords that might be relevant here: Perron-Frobenius theory, Pisot number, Salem number. I'll also mention that apparently these ideas are used in the theory of quantum groups.

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1  
I should also mention, because it's not clear to me whether you know this or not, that you can turn this into a formula which expresses the coefficients of a polynomial whose roots are u_i v_j or u_i + v_j in terms of the coefficients of polynomials whose roots are u_i and v_j, respectively, by taking the matrices to be companion matrices: en.wikipedia.org/wiki/Companion_matrix –  Qiaochu Yuan Apr 15 '10 at 15:42

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