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In Uniswap v3, you can have up to three fee-designated pools (0.05%, 0.30% or 1.00%) per token pair.

Do the token prices remain exactly the same across these pools or are there discrepancies across ticks? If not, this would open up arbitrage opportunities across these pools, no?

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token prices do not remain exactly. the relative price of those tokens in the pool is determined by the constant product formula (k). This "k" will be constant throughout the pool duration.

Assume you have A and B tokens in a pool. If the amount of A token decreases to keep the "k" value constant its price must increase. (Basic finance rule. scarcity increases the price). And yes this will create an arbitrage opportunity which will incentivize the others to bring more A token to the pool which will lead to price balance again.

In uniswapv3, those amounts are not checked. Relative price is

 1.0001 ^ Tick 

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