The link stopped to work, I instead give now the title of the article.

Do you mean $\sum_{i=1}^m x_i^3\ge a_3$ instead of $\sum_{i=3}^m x_i^3\ge a_3$? If yes, do you mean $\sum_{i=1}^m x_i^3 = a_3$ instead of $\sum_{i13}^m x_i^3 \ge a_3$?

I guess it is not clear whether it is really doable in polynomial time. The complexity status of the semidefinite feasibility theorem is unknown. Although some people believe it could be in P it is not even known to be in NP.

You want the family to be a continuous path, of course. An easy observation: Call the first polynomial f, the second g. The answer is yes if f and g differ only in the coefficient of z^k: Then z f' - k f interlaces f in the sense of Definition 4.1 in [arxiv.org/pdf/1304.4132.pdf] (this uses that f has no positive roots). So does z g' - k g interlace g. But z f' - k f = z g' - k g. Hence f and g have a common interlacing. Now apply Lemma 4.5 in [arxiv.org/pdf/1304.4132.pdf].