Disparity between polynomials and integers Many results in number theory and algebraic geometry have an equivalent (or almost equivalent) result in integers and polynomial rings. It is also common to use techniques from one area to the other and prove results that resisted solution for a long time. In fact, we have very well known mathematical analogies between integers and ring of polynomials. 
However, there are still problems that are easy on ring of polynomials and yet seem to be hard on integers. For example, factoring a polynomial is easy while factoring over the integer is one of the hardness assumption. Similarly, the polynomial variant of finding shortest vector for lattices is easy while it is hard over a lattice over integers. Again the polynomial variant of prime number theorem if proven in the case of integer will actually give a proof of Riemann Hypothesis. This list goes on and on. 
I often wonder what is the intuitive reason for this disparity? Is it the case that the polynomial variant of a problem is always easy than that over the integers, or is there no such hardness reduction?
 A: As Paŭlo alludes to, it's hard to answer this carefully without knowing exactly what part of the analogy you're referring to.  So this will be a pretty rough response.
First, there's a very surface-level (but still important) analogy between, say, $\mathbb{Z}$ and $\mathbb{C}[x]$, most of which stems from the fact that these two rings both have division algorithms.  You get a Euclidean algorithm, gcd's, principal ideal theorem, etc., all of which lead to nice analogies: Lagrange interpolation vs. standard Chinese remainder theorem, Taylor series vs. $p$-adic expansions, etc.  In this setting, probably the biggest reason that jumps to mind for the disparity you mention is the local phenomena.  First of all, it's easier to add series in $\mathbb{C}[[x]]$ than it is in $\mathbb{Z}_p$: Taylor series add coefficient-by-coefficient, adding $p$-adic expansion involves "carries." Really what this is pointing to is the degree function being a significant boon in the polynomial case, being somewhat more rigid than $p$-adic valuation.  Second, the fraction field $\mathbb{Q}$ of $\mathbb{Z}$ has a very weird completion that satisfies, of all things, the Archimedean property. For contrast, all the completions of $\mathbb{C}(x)$ are non-Archimedean and all of their residue fields are identical. You could even re-phrase this topologically -- $\mathbb{Q}$ has a connected completion, $\mathbb{C}(x)$ does not.
There is a stronger analogy between number fields and function fields over finite fields. Here your residue fields are finite fields like they are in $\mathbb{Z}$, hence the tightening of the analogy.  This an area which lies under much of algebraic geometry, and is too bewilderingly vast to say much about here. Let me point you to a couple of MO questions that address this point:


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*https://mathoverflow.net/questions/1367/global-fields-what-exactly-is-the-analogy-between-number-fields-and-function-fie

*https://mathoverflow.net/questions/41770/explaining-the-number-field-function-field-analogy
Suffice it to say that the full force of algebraic geometry can come to bear on the function field side of things, so we get Weil conjectures, ABC, Riemann hypothesis for function fields, etc. It's probably worth mentioning that there are still plenty of instances where we don't know either of the analogous results (e.g., Artin's primitive root conjecture vs. Lang-Trotter, etc.)
Finally, let me mention (very) briefly a third level to this analogy, which is to introduce the notion of a Drinfeld module, designed (in a sense) to tighten once again the analogy. Again this is too gigantic a subject to say anything meaningful about in a paragraph, but I mention it as a point where the disparity starts to disappear and/or reverse — there are open questions about Drinfeld modules whose arithmetic analogues are known. Perhaps the commenters can think of an example that doesn't require pages of notation to write down.
