10
$\begingroup$

complex plot of the zeros

(Diagram and setup from UCSMP Precaluclus and Discrete Mathematics, 3rd ed.)

Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Are the curves in the locus some specific (named) type of curve? Is there a simple way to describe the curves (equations)?

The question need not be limited to the specific polynomial given--a similar sort of locus is generated by the zeros of nearly any quartic polynomial as the constant term is varied.

  • 2
    Wouldn't this be spec(Z[x,y]/(xP(x)+y))(C) for a cubic polynomial P(x)? (Where we view the scheme spec(...)(-) as a functor CRing^op->Sets).2010-07-23
  • 0
    (I can't answer Harry's question as category theory is where abstract algebra stopped making any sense to me.)2010-07-23
  • 0
    Do you expect to do any better than what the quartic formula gets you?2010-07-29
  • 0
    @Qiaochu Yuan: Not expect, but hope.2010-07-29
  • 0
    @Isaac: Is it just me, or is the image that is supposed to appear in your question missing?2010-12-11
  • 0
    @Jonas Meyer: It was not just you. This post was from before SE supported direct image uploads and the image host I'd used unexpectedly deleted out some images. I thought I'd fixed them all, but apparently I missed this one. It should be fixed now.2010-12-11

4 Answers 4