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Working in the Blum-Shub-Smale model of computation on the real numbers, weanswer several questions of Meer and Ziegler. First, we show that, for eachnatural number d, an oracle for the set of algebraic real numbers of degree atmost d is insufficient to allow an oracle BSS-machine to decide membership inthe set of algebraic numbers of degree d + 1. We add a number of furtherresults on relative computability of these sets and their unions. Then we showthat the halting problem for BSS-computation is not decidable below anycountable oracle set, and give a more specific condition, related to thecardinalities of the sets, necessary for relative BSS-computability. Most ofour results involve the technique of using as input a tuple of real numberswhich is algebraically independent over both the parameters and the oracle ofthe machine.

Noncomputable functions in the Blum-Shub-Smale model