# All integer polynomials of the form $x^{d_t}-x^{d_1-1}-x^{d_2-1}…x^{d_t-d_t}$ with the same maximal (real) root.

Let $$d=(d_1,\dots,d_{d})$$ be a vector of integer exponents of a polynomials such that $$d_1\leq d_2\leq \dots \leq d_{|d|}$$. Let us define the polynomial $$p_d(x)=x^{d_{|d|}}-\sum_i{x^{d_{|d|}-d_i}}$$ (such that $$1-\sum_i{x^{-d_i}}=0$$ iff $$p_d(x)=0$$). These are all polynomials where only the power of the largest exponent has a positive coefficient.

I am interested in every polynomial $$p_{d'}(x)$$ (or vector of integers $$d'$$) of the same form such that the largest real root $$r$$ of $$p_{d'}(x)$$ and $$p_{d}(x)$$ is the same. For example $$x^{1}-2\cdot 1$$, $$x^{2}-4\cdot 1$$, $$x^{3}-8\cdot 1,\,\dots$$, with the root $$x=2$$, or $$x^{2}-x^{1}-1,x^{4}-x^{1}-x^{2}-x^{2}-1,\dots$$ with the (real) roots $$\frac{1\pm \sqrt{5}}{2}$$.

I already know that we can get another vector $$d'$$ of exponents from a known one: Just add all combinations of the exponents: From $$(2,1)$$ we can get $$(4,3,3,2)$$. The real roots of the related polynomial should be the same then. However, it is not clear for me, if a) there can be multiple vectors of the same length, and b) if there are other vectors possible that cannot be produced by this method.

• For example, all the polynomials $p(x) = x^m - x^k$ for $k < m$ have largest real root $1$. – Robert Israel Jan 10 at 16:11
• That's right, but unfortunately these polynomials are not what I mean. There should be $−x^{d_{|d|}−d_{|d|}}=x^0$ in the term. – ptid91 Jan 11 at 9:51

$$x^7-x^5-x^4-x^2-1$$ and $$x^7-x^6-x^2-x-1$$ both have the same largest real root, namely the real root of $$x^3-x^2-1$$ which is a factor of both.