# Valuation of polynomials

Mathematics Asked by Muselive on November 27, 2020

My paper defines a valuation on a Ring $$R$$ to be a map $$v:R-{0} rightarrow A$$ where $$A$$ is an ordered abelian group. This map has the following properties;

1.$$v(ab)=v(a)+v(b)$$

2.$$v(a+b)geq min{v(a),v(b)}$$
(note this is missing the usual conditon wich grants equality if $$v(a) neq v(b)$$)

My question is the following: Let $$w:R[x]-0_{R[x]} rightarrow A$$ by;
$$w(sum_{i=0}^nr_ix^i)=min_{0leq i leq n}v(r_i)$$

and I’m to show this is a valuation. I can easily show property 2 but I cannot show property 1. I’m starting to doubt if it is possible; I wrote out some examples and I don’t see how I can get anything other than a lower bound for a $$w(fg)$$. A hint or even confirmation that this is solvable would be very appreciated.

If $$R$$ is an integral domain, your formula for $$w$$ does yield a valuation. Here is a hint toward proving condition 1 holds. We will use the observation in my comment that $$v(a+b)=min{v(a),v(b)}$$ if $$v(a)ne v(b)$$.

Let $$f=sum_i r_ix^i$$ and $$g=sum_j s_jx^j$$ and suppose $$w(f)=v(r_p)$$, $$w(g)=v(s_q)$$, where $$p,q$$ are as small as possible. In $$fg$$, the coefficient $$t_{p+q}$$ of $$x^{p+q}$$ is a sum containing the product $$r_ps_q$$ and other products $$r_is_j$$ where either $$i or $$j. Show $$v(r_is_j)>v(r_ps_q)$$ for those terms, and so $$v(t_{p+q})=v(r_ps_q)=v(r_p)+v(s_q)$$. Show that no coefficient of another $$t^k$$ can be smaller than $$v(r_p)+v(s_q)$$. Conclude that $$w(fg)=v(r_p)+v(s_q)=w(f)+w(g)$$.

Correct answer by Allen Bell on November 27, 2020

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