When you use arcane surge the target has to roll his saving throw twice and take the lower result. If you are making a caster level check to overcome spell resistance you get to roll twice and take the better result. So fare everything is pretty clear.
Conviction only allows you to reroll one of your own rolls, so you cannot use it to force the target to reroll their saving throw. You can use it to reroll your caster level check to overcome spell resistance. If you do you throw out the result of the first roll completely and reroll it twice and take the higher of the two rolls you just made. Since you have control over when you use it you will only be using it when you roll poorly.
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Conviction doesn't work on Caster Level checks:
Source
PRPG Core Rulebook pg. 74
Conviction (Su): At 15th level, you can reroll any one ability check, attack roll, skill check, or saving throw you just made. You must decide to use this ability after the die is rolled, but before the results are revealed by the GM. You must take the second result, even if it is worse. You can use this ability once per day.
An 'Ability Check' is defined as a check "using just an ability score" like a strength check to force a door open, so a caster level check to pierce spell resistance does not fall into any of these categories.
Even if this did work, I don't believe there is any consensus on how mixing re-rolls with 'roll x and take the best' works, so it would be completely up to your GM...
Actually Taja is correct in that Conviction does not allows a reroll of a caster check.
If it did you would have to decide after you rolled, but before you the GM announced if it succeeded. So this would only be useful if you botched both rolls. If you choose to use Conviction than the original roll(s) are discarded as if they never happened and you roll again. Since the spell was still cast with arcane surge all of its effects would still be in place including rolling twice and adding your tier to the result.
Conviction is only going to be useful in situations where you have an obvious failure. Otherwise it is possible that you reroll a success and replace it with a failure. In keeping with the spirt of the ability a GM should not be telling the player what they need to roll before they roll in these cases.
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fine Fortunate charm then, conviction was just the reroll power i had most recently seen, and yes we allow it to be used on SR checksConviction doesn't work on Caster Level checks:
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PRPG Core Rulebook pg. 74Source
Conviction (Su): At 15th level, you can reroll any one ability check, attack roll, skill check, or saving throw you just made. You must decide to use this ability after the die is rolled, but before the results are revealed by the GM. You must take the second result, even if it is worse. You can use this ability once per day.An 'Ability Check' is defined as a check "using just an ability score" like a strength check to force a door open, so a caster level check to pierce spell resistance does not fall into any of these categories.
Even if this did work, I don't believe there is any consensus on how mixing re-rolls with 'roll x and take the best' works, so it would be completely up to your GM...
As I said, there is no actual rule or even a common consensus on how this interaction should work: Your table and/or GM will have to determine this for themselves.
On a side note, if you are this concerned about SR on a mythic sorcerer, might I ask if you have heard the gospel of Desna and her Starlit Caster boon: Source
Inner Sea Gods pg. 44
2: Starlit Caster (Su) Over time you have learned to focus your magical power to better damage agents of evil. You add your Charisma bonus on your concentration checks,* as well as on your caster level checks to overcome spell resistance. In addition, when you stand in starlight and cast a spell that deals hit point damage, you can have it deal an extra 2d6 points of damage. This bonus damage is untyped, and manifests as a glowing aura of starlight around the spell’s original effect.
Sorcerer is a natural match for becoming an Evangelist of Desna, or you could take the Deific Obedience + Diverse Obedience feat combination...
*No, I have no idea how the 'add charisma bonus to concentration checks' is supposed to work with classes that already use charisma: I believe this ability pre-dates the 'no using the same untyped stat bonus twice' rule and I'm not aware of any clarification on it: Personally, I figured it should probably be a 'Sacred' bonus instead of untyped, but that's just me...
Since there is not going to be anything else written on1 edition by Paizo there will be no rule clarification. So far two of the three people on this post are agreeing how it works and the third is abstaining from posting any opinion on how it works. 2 out of 3 is a pretty clear majority in favor of two rolls.
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Since there is not going to be anything else written on1 edition by Paizo there will be no rule clarification. So far two of the three people on this post are agreeing how it works and the third is abstaining from posting any opinion on how it works. 2 out of 3 is a pretty clear majority in favor of two rolls.
Well, one of those 3 is the OP who asked the question, so really the responses have been one person who thinks multiple rolls and one person who says you'll have to decide for yourself.
In no case can you actually draw any real conclusions from a sample size this small...
That answer pretty much cover almost any rules question since 2nd edition came out. Since we are not going to get a developer to weigh in on the question How would you rule in a game you ran?