Why do antinuetrinos get emitted during beta decay? Why do antinuetrinos get emitted, during Beta Decay? Can the existence of them be discovered mathematically (and if so, how?), or is their existence only proven, via empirical means?
 A: This is because of a principle known as lepton number conservation. "Leptons" are a class of elementary particles which includes both electrons and neutrinos (the only stable examples), as well as their associated anti-particles. The lepton number is the total count of leptons in a physical system, where that anti-leptons are counted as though they were a "negative number of leptons", so two anti-leptons is the same as -2 leptons, or a lepton number of -2. Having a lepton and an anti-lepton together is, in this sense, equivalent to having 0 leptons.
This principle says that the total number of leptons, defined thus, does not change, similar to how that conservation of energy says that a system's total energy does not change, and conservation of momentum says that a system's total momentum does not change.
In a beta decay of the type you mention, an electron counts as one lepton. This electron did not exist before - it is created in the decay. Because we are creating a lepton, the lepton number would, if nothing else were emitted, go up by 1. This would violate the principle. What you see at work here is similar to how Newton's third law works: think about a rocket. For a rocket to move forward (accelerate), it has to create forward momentum. But because of conservation of momentum, the total system momentum cannot change, and hence the forward momentum must be balanced out by a counteracting backward momentum, which is why that rockets must eject propellant.
Likewise, when the beta decay creates a new electron, some other kind of anti-lepton (which counts for -1) must be produced, and given the energies involved, the easiest to produce is an anti-neutrino, which has negligible (but not zero!) mass, and hence a negligible minimum energy cost (via the famous relationship by Einstein, $E_\mathrm{rest} = mc^2$). The sum total number of leptons created is then 1 + -1 = 0, which meets the conservation requirement, just as with our rocket example.
(You may ask: why not emit an electron and an anti-electron, i.e. a positron, instead? That would still conserve lepton number. The answer to that is charge conservation - and the same principle goes: a proton has a charge of +1 e [$+1.602 \times 10^{-19}\ \mathrm{C}$], and to convert that to a neutron, that charge must be neutralized. Emitting an anti-electron, with also a charge of +1 e, would cancel the electron's -1 e charge, and then that would mean the proton's charge would have to have vanished - a conservation violation. So a neutral antiparticle must be emitted instead, and the only decent choice is the antineutrino. All relevant conservation laws must be upheld for a decay or other process to be permitted.)
Insofar as "discovering their existence mathematically", no. They are an observed feature of our Universe or, at least, depending on one's philosophical tack, our most useful way of understanding its basic processes is best described by including them. That said, the neutrino was actually hypothesized based on observations of these decays, but for a different reason: conservation of energy, since before its discovery, the only energy that could be seen to be emitted was that in the electron (or positron), but this left a significant deficit, and hence it was proposed that there must be another particle being emitted. That "other particle" would later be found to be the (anti-)neutrino, which eluded easy detection due to it being notoriously poor at interacting with anything (though much less stubbornly non-interactive than Dark Matter). The just-mentioned lepton number conservation principle was then ascertained after finding this out.
