On the Walach - Zeheb multivariable positivity test

In a recent article in this TRANSACTIONS Walach and Zeheb presented a test for the positivity of real multivariable polynomials on rectangular domains. While the basic idea of the Walach-Zeheb test is sound and potentially useful, the case of unbounded domains is in need of reexamination, and, what is more important, the restriction to rectangular domains can be dispensed with entirely. Using the Fritz John Multiplier Rule, the present note extends the scope of the Walach-Zeheb test to include arbitrary closed domains, and clarifies what is entailed in the case of unbounded domains. In particular it is shown that the test actually tests for strong positivity, i.e., for the existence of positive lower bounds.