# Polynomials that share at least one root

This is a generalization of an MSE question, Polynomials that share at least one root.

Let $$P(x)$$ be a specific polynomial of degree $$d$$, with given real coefficients $$A_i$$ ($$A_d=1$$), and real roots: $$P(x) = x^d + A_{d-1}x^{d-1} + A_{d-2}x^{d-2} + \cdots + A_{0}\;.$$

Q. What does the set of polynomials, with real coefficients $$a_i$$, $$p(x) = x^d + a_{d-1}x^{d-1} + a_{d-2}x^{d-2} + \cdots + a_{0} \;,$$ look like (geometrically) in $$\mathbb{R}^d$$, when each $$p(x)$$ shares at least one root with $$P(x)$$?

I am seeking a description of this set in the space of the $$d$$ coefficients, $$(a_0,\ldots,a_{d-1})$$. The reason there is hope for a nice description, is that it makes a pretty picture for $$d=2$$.

Let $$P(x) = x^2 + A_1x + A_0$$, with $$A_1=3$$ and $$A_0=-1$$. The plot in the $$a_0 a_1$$-plane of all other $$p(x)= x^2 + a_1x + a_0$$ that share a root with $$P(x)$$, looks like this:

Lines intersect at $$(a_0,a_1)=(-1,3)$$. The discriminant is $$a_1^2 = 4 a_0$$.
All those $$(a_0,a_1)$$ on the two lines share a root with $$x^2 + 3x -1$$. The lines are tangent to the discriminant parabola.

• Resultant of these polynomials must vanish. So it is an algebraic surface with concrete equation (see en.wikipedia.org/wiki/Resultant ) Commented Jul 1, 2019 at 23:39

The locus of real polynomials $$p(x)$$ sharing a root with $$P(x)$$ is the union of the hyperplanes $$H_\alpha : p(\alpha)=0$$, where $$\alpha$$ runs over the roots of $$P(x)$$. This is an arrangement of hyperplanes.
The intersection of $$H_\alpha$$ with the discriminant variety $$D : \mathrm{disc}(p)=0$$ contains the subspace $$H^{(1)}_{\alpha} : p(\alpha)=p'(\alpha)=0$$. In fact $$H_\alpha$$ and $$D$$ are tangent along $$H^{(1)}_\alpha$$. To see this, take a generic polynomial $$p_0(x)=(x-\alpha)^2 q_0(x)$$ in $$H^{(1)}_\alpha$$. Near $$p_0$$, polynomials in $$D$$ will be of the form $$p(x)=(x-\alpha+\varepsilon)^2 q(x)$$ where $$q$$ is near $$q_0$$. Then $$p(\alpha)=\varepsilon^2 q(\alpha)$$, while the distance from $$p_0$$ to $$p$$ is of first order with respect to $$\varepsilon$$.
I tried to make a 3D image for $$P(x)=x^3+3 x^2-2 x-1$$. The set consists of three planes, each tangent to the discriminant surface. But it became too visually complex, partly because the discriminant is complicated. For what it's worth: