With this article, we hope to launch the investigation of what we call the
real zero amalgamation problem. Whenever a polynomial arises from another
polynomial by substituting zero for some of its variables, we call the second
polynomial an extension of the first one. The real zero amalgamation problem
asks when two (multivariate real) polynomials have a common extension (called
amalgam) that is a real zero polynomial. We show that the obvious necessary
conditions are not sufficient. Our counterexample is derived in several steps
from a counterexample to amalgamation of matroids by Poljak and Turz\'ik. On
the positive side, we show that even a degree-preserving amalgamation is
possible in three very special cases with three completely different
techniques. Finally, we conjecture that amalgamation is always possible in the
case of two shared variables. The analogue in matroid theory is true by another
work of Poljak and Turz\'ik. This would imply a very weak form of the
Generalized Lax Conjecture.Comment: 24 page