How does the gold/gem exchange rate get determined?

You can pay real money to purchase gems, which introduces new gems to the system and balances out those gems spent on gem store purchases. If the rate at which gems are spent outpaces the rate at which they are bought, the total number of gems in circulation will decrease, and the gold:gem ratio will increase until it reaches a point at which people are more willing to spend real money on gems than to pay the exorbitant gold prices for gems. In this manner a balance would be maintained in the number of gems in the system.

It's a pseudo economy towards Anet's favor. The graph under gem is the average/lowest/highest purchase price for 100 gems. Basically it works like this, you trade 2 apples for 4 oranges but you can only always trade 4 oranges for < 2 apples. See how that works? In real economy the exchange rate of apples/oranges can differ and fluctuate where sometimes apple is higher and sometimes orange is. However, in GW2's economy the apples (in-game gold) will always be higher than oranges (gems), i.e. if you trade 10 silvers for 39 gems, 39 gems will always trade for < 10 silvers. The exchange rate may vary but the relationship will not, for example, at some point it may be 10 silvers for 35 gems instead of 39 but 35 gems will always trade for < 10 silvers. There's really no point of keeping track because you can never win, I'd say 10 silvers for > 37 gems would be a good deal.

Furthermore, this is what I think how the gold/gem exchange rate is set: depends on how many people give money to Anet for gem and use that gem to buy gold. Ultimately Anet is a for profit company, the economy has to work towards their advantage. Regardless, it is still better than a lot of other game, at least it allows you to not spend real money but get the same benefit by investing time.

It is my theory that ArenaNet is directly controlling the values.

In the normal buy sell economy, a player has a choice when selling or buying an item. They can choose to take the offer made available to them, or they can make their own better counter offer. The Gold/Gem exchange does not offer the option to make a better deal, it is directly forcing you to accept the value the company is placing on gems.

From what I can tell, there is no player interaction involved. It's simply you buying and selling to Anet. They likely have a complex program that tracks the number of gems purchased for $ and how many of those gems are being traded for gold and bases prices on this model with a substantial markup on players who are trading gold back into gems.