Here is a partial answer. I believe for `Method -> {opts}`

, `opts`

may be any of the following, and they will have whatever effect they have:

```
Internal`InequalitySolverOptions[]
Internal`ReduceOptions[]
Internal`NSolveOptions[]
(*
{"ARSDecision" -> False, "BrownProjection" -> True, "CAD" -> True,
"CADAlgebraicCoefficients" -> True, "CADBacksubstitution" -> Automatic,
"CADCombineCells" -> True, "CADConstruction" -> Automatic,
"CADDefaultPrecision" -> 30.103, "CADExtraPrecision" -> 30.103, "CADMethod" -> Automatic,
"CADNRootsMethod" -> Automatic, "CADSortVariables" -> Automatic,
"CADZeroTest" -> {0, ∞}, "EquationalConstraintsCAD" -> Automatic,
"FGLMBasisConversion" -> False, "FGLMElimination" -> Automatic,
"GenericCAD" -> True, "GroebnerCAD" -> True,
"LinearDecisionMethodCrossovers" -> {0, 30, 20, Automatic},
"LinearEquations" -> True, "LinearQE" -> True, "LWDecision" -> True,
"LWPreprocessor" -> Automatic, "ProjectAlgebraic" -> Automatic,
"ProveMultiplicities" -> True, "QuadraticQE" -> Automatic,
"QVSPreprocessor" -> False, "ReducePowers" -> Automatic,
"RootReduced" -> False, "Simplex" -> True,
"SimplifyInequalities" -> Automatic, "ThreadOr" -> True, "ZengDecision" -> False}
{"ADDSolveBound" -> 8, "AlgebraicNumberOutput" -> True,
"BDDEliminate" -> Automatic, "BooleanInstanceMethod" -> Automatic,
"BranchLinearDiophantine" -> False, "CacheReduceResults" -> Automatic,
"DiscreteSolutionBound" -> 10, "ExhaustiveSearchMaxPoints" -> {1000, 10000},
"FactorEquations" -> Automatic, "FactorInequalities" -> False,
"FinitePrecisionGB" -> False, "ImplicitIntegerSolutions" -> Automatic,
"IntervalRootsOptions" -> {"AllowIncomplete" -> True, "FailDepth" -> 20,
"MaxDepth" -> 50, "MaxFailures" -> 100, "MaxIncomplete" -> 1000,
"MaxSimplified" -> 1000, "MaxSteps" -> 100000, "MinPrecision" -> 12},
"LatticeReduceDiophantine" -> True, "LinearEliminationMaxDepth" -> ∞,
"MaxFrobeniusGraph" -> 1000000, "MaxModularPoints" -> 1000000,
"MaxModularRoots" -> 1000000, "MaxPrimeIndex" -> 1000000000,
"NIntegrateTimeConstraint" -> 60, "PresburgerQE" -> Automatic,
"QuickReduce" -> False, "RandomInstances" -> Automatic,
"RealRootsOptions" -> {"MaxExtensionDegree" -> 5, "MaxFactorSquareFreeDegree" -> 10000,
"MaxNestedRootsDegree" -> 100, "SparsityThreshold" -> 0.02},
"ReorderVariables" -> Automatic, "SieveMaxPoints" -> {10000, 1000000},
"SolveDiscreteSolutionBound" -> 1000000, "SyntacticSolveAssumptions" -> False,
"TranscendentalRecursionLimit" -> 12, "UseNestedRoots" -> Automatic,
"UseOldReduce" -> False, "UseTranscendentalRoots" -> Automatic,
"UseTranscendentalSolve" -> True}
{"ComplexEquationMethod" -> Automatic, "MonomialOrder" -> Automatic,
"ReorderVariables" -> True, "SelectCriterion" -> (True &),
"Tolerance" -> 0, "UseSlicingHyperplanes" -> True}
*)
```

I do not know if there are other settings that may be used. Building on comments by belisarius and Mr. Wizard, many (or maybe all) of the above are `SystemOptions`

that may be passed via the `Method`

option to `FindInstance`

. I do not know what all these options do. Here is an example to show they, or at least one of them, have an effect when passed through the `Method`

option (see the discussion of the `"SieveMaxPoints"`

option in Diophantine Polynomial Systems for an example that was the basis for this one):

```
FindInstance[x^2 + 21 y^3 - 17 z^4 == 632, {x, y, z}, Integers]
```

FindInstance::nsmet: The methods available to FindInstance are insufficient to find the requested instances or prove they do not exist. >>

```
FindInstance[x^2 + 21 y^3 - 17 z^4 == 632, {x, y, z}, Integers,
Method -> {"SieveMaxPoints" -> {10^4, 10^8}}]
(* {{x -> -944, y -> 22, z -> 16}} *)
```

**Update - another example.** This one is from Real Polynomial Systems mentioned by @belisarius. It is the last example in the tutorial and it is supposed to show a case in which `"ZengDecision" -> True`

improves the timing. In that respect things seem to have changed, but the option still has an effect.

```
FindInstance[
x^4 + y^4 + z^4 + w^4 - 5 x y z w + x^2 + y^2 + z^2 + w^2 + 1 < 0,
{x, y, z, w}, Reals] // AbsoluteTiming
(* {0.140519, {{x -> -6, y -> -5, z -> -6, w -> -4}}} *)
FindInstance[
x^4 + y^4 + z^4 + w^4 - 5 x y z w + x^2 + y^2 + z^2 + w^2 + 1 < 0,
{x, y, z, w}, Reals, Method -> "ZengDecision" -> True] // AbsoluteTiming
(* {0.4111, {{x -> -(7/2), y -> -4, z -> -3, w -> -3}}} *)
```

Note: There are other `Internal`*Options`

with corresponding `Internal`Set*Options`

, which might be used in certain cases of `FindInstance`

.
And if not in `FindInstance`

, then perhaps they can be used in other functions with an obscure `Method`

option. Each with one exception corresponds to a group of `SystemOptions[]`

.

A perusal of

`??System`InstanceDump`*`

might provide a few clues, I think. – J. M.'s ennui – 2012-06-14T12:43:49.000Related> Here there are a few options for

– Dr. belisarius – 2012-06-14T12:59:56.267`FindInstance[]`

mentioned http://reference.wolfram.com/mathematica/tutorial/RealPolynomialSystems.html@J.M. that command yields

`Information::nomatch: No symbol matching System`InstanceDump`* found. >>`

on my system. – Mr.Wizard – 2012-06-14T13:13:02.083@belisarius I see the low-level (and wide effect)

`SystemOptions`

, but not`Method`

options. Am I overlooking something? – Mr.Wizard – 2012-06-14T13:13:49.773Ech, I keep forgetting that you're on version seven... – J. M.'s ennui – 2012-06-14T13:14:38.603

@Mr.Wizard No, you are right. That is the reason why I didn't add it as an answer and started my comment as "Related". I thought the info in the article could be of your interest, though. – Dr. belisarius – 2012-06-14T13:19:23.427