# Explanation for certain options of SolveAlways?

As one can see,

In[1]:= Options[SolveAlways]

Out[1]= {InverseFunctions -> Automatic, MakeRules -> False,
Method -> 3, Mode -> Generic, Sort -> True,
WorkingPrecision -> \[Infinity]}


Both InverseFunctions -> Automatic and WorkingPrecision -> Infinity are well-documented; MakeRules -> True has been explained here, Mode -> Modular is explained here, and Sort -> True has been explained here. But what are the meanings of Method -> 3 and Mode -> Generic?
According to Message[SolveAlways::method, "1"],

Value of option Method -> 1 should be 1, 2, or 3.

And according to ?Mode,

Mode is an option to Solve and related functions that specifies in what sense the equations are to be solved. The possible settings for Mode are Generic, Modular, and Rational.

So, what is the difference among the three Method options and the three Mode options?

• These are outdated options. Legacy from an earlier time. Commented Dec 17, 2023 at 16:25
• @DanielLichtblau Thanks. Yet is the Mode also out-dated? The help page does not mark it as an out-dated option. Besides, in Options[SolveAlways], the default method is Method -> 3, while in Options[Eliminate], the default method is Method -> 1; is there any difference? Commented Dec 18, 2023 at 6:27
• Those are settings I have not touched in this millennium. Are the Method ones different? Probably. What do they do that's different? I don't remember. Re Mode, I believe the idea was that one could use the Generic to get Solve to give generic solutions ad Reduce give a full breakdown using the Rational setting (maybe). One could use Mode->Modular and give an extra equation, and Solve would try to find a modulus for which the system was noy over-determined. My guess is this can instead be handled with an existential quantifier, in a way that might be more reliable. Commented Dec 18, 2023 at 15:55
• I will add that I do not see delving into this legacy design as a particularly productive use of time. It's been superceded for a reason. Commented Dec 18, 2023 at 15:56