Skip to main content
1 of 2
David Feldman
  • 17.6k
  • 8
  • 67
  • 135

Nullstellensatz for quaternionic plane curves?

By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after identifying ${\Bbb H}$ with ${\Bbb R^4}$, one can capture with a single quaternionic polynomial any locus that occurs as the set of solutions of four real polynomials. (One needs a trick: if $Q=a+bi+cj+dk$, $(-Q+iQi+jQj+kQk)/2=a$; and similarly for $b$,$c$ and $d$).

Is there a criterion in terms of ${\Bbb H}<x,y>$ for two polynomials to cut out the same zero locus? Of course in principle one could unwind everything and then appeal to some appropriate realnullstellensatz, but I'm interested in a solution that is idiomatically quaternionic.

And a softer question. Capturing a real 4-fold in ${\Bbb R}^4$ as a specific quaternionic plane curve singles out a way to produce the 4-fold as the intersection of 4-hypersurfaces. That imposes a great deal of extra structure on the 4-fold. Indeed presumably too much, seeing as certain natural quaternionic manipulations (multiplication by non-zero constants on either side) change the intersecting polynomials but not the locus. What is the right level of extra structure in order to move to the quaternionic picture?

David Feldman
  • 17.6k
  • 8
  • 67
  • 135