The GPS signal is already under the noise floor on Earth there is no way with present day tech that you'd be able to get the signals on Mars and besides that the math would not work because the round-off errors regarding the timing would make all satellites appear at the same point in space when looked at from Mars. The distance is hard to appreciate in terrestrial terms without getting into 'golf balls and oranges' kind of explanations.
> round-off errors regarding the timing would make all satellites appear at the same point in space when looked at from Mars
Not just timing, also space - Earth is a fraction of an arc second on Martian sky, pretty much a very small dot. All your signals would be coming from that small dot, i.e. practically on top of one another.
The whole idea of GPS is to turn a very good time reference (an atomic clock) into a space reference, by using multiple such references with known locations at once and then to send out time stamped signals. If you can't distinguish arrival times the signal origins collapse into a point. The travel time of the signal from Earth to Mars is anywhere from 4 to 24 minutes (assuming you have line of sight, which isn't the case when Mars is on the other side of the sun), with flight times that long the difference in arrival time between the signals is meaningless.
The difference in position between the satellites is at best 40,000 km (MEO orbits and satellites in opposite positions around Earth would give you the largest baseline), the difference from Mars to Earth is > 50M kilometers. So when viewed from Mars this would be like trying to triangulate your position in the United States based on signal sources spaced a very short distance apart somewhere in Moscow.
(sorry for the strained analogy)
Another problem with the approach is that of the two possible solutions that are the result from computing your location from GPS satellites one of the solutions is deep inside the Earth, which for both radio related reasons and reasons of practicality can be safely ignored. From a location in space very far away that trick no longer works so you will end up having to pick one of several answers.
Of course you could stick a GPS like transmitter into every Mars orbiter we launch from now on so that at some point there will be enough coverage locally to allow navigation, but that's a pretty expensive trick, besides that you'd also need a bunch of base stations in order to properly compute the orbits of the satellites to the required precision so that you can tell the satellites where they are.
I'm pretty sure if you're on the Earth's surface and can pick up three GPS satellites, the other solution is out in space. If you see four or more GPS satellites, there's a unique solution.
That's probably correct. I read up on GPS years ago and I remember that one of the possible solutions is intuitively wrong for a surface of the earth measurement and filled that in mentally as 'inside the earth', but out in space sounds much more logical and likely.
