Statement of purpose: How to state PI backup I am a foreign Biology student applying to USA phd programs.
I have contacted several PIs asking if they had an available position (because they work within my interests). I've got a lot of no responses, some &quot;No funding/no position&quot; and I've got some positive feedback. I have one particular university in which 3 PIs sent me back an &quot;encourage to apply&quot; mail.
Examples of these are:

You seem like a good fit (...)
I'll keep an eye out for your application
I might have some influence on comitee X
If you get accepted (I have no influence) ask to chat with me

Or just &quot;I encourage you to apply&quot;.
From my point of view, the universities take into account that the PIs are willing to accept the candidate (they say the measure is &quot;fit&quot; not only grades). Then it would be more than wise to state clearly in my statement of purpose that I've spoken with Professor X about working there and received positive feedback. However, it might seem pedantic to state that &quot;Professor X...&quot;. Just to make things clear, I'm not an outstanding, perfect grades, so I believe this could be a very important feature of my statement.
Finally, it is also stated clearly on many programs that PIs have no influence on comitee decisions. I believe that there's some &quot;legal hole&quot; in which they don't want to give control to PIs but they find themselves needing PI's approval or something.
My question is: Should I include this information? How to do it politely?
Let's just clear the situation within the scope of two cases:


The Pl-based universities: In this case, acquisition of a positive response from a Pl would pave your way, considerably. Within this scenario, you are supposed to mention your propensity toward the target Pl within SoP. Then, your SoP will be analyzed and sent to that fellow, has which been named within that SoP. Here is the very step, within which the Pl could admit you.

You better to pay close attention to two important points: 
interpretation of the Pl's tone within the response is, merely, up to yourself... For example, he/she might say that: "You look a qualified applicant, I will encourage you to apply"..., but the other one's reply might be: "Your CV sounds considerably overlapping with my current projects. Would you please take some explanation into account regarding your programming skills, as most the current projects are highly-simulated-based ones...". One could claim that the second answer might be more promising than the first one, but the others may take both energizing and apply into both... So, you need to apply for the replies, are which stemmed from serious propensity of the corresponding Pls. Hence, you must cite the desired Pl's name within the application.
The admission-committee-based universities: In this case, the Pl's reply just could be effective to find your turning point. Actually, their positive opinion could be compiled as your gratifying situation in comparison with the typical admitted students to the program. In contrast, the deferring answers should warn you about the high risk of the taking application into account. Therefore, mentioning the Pl's names in this scenario is not useful, noticeably.