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# Resolve

 Resolve[expr]attempts to resolve expr into a form that eliminates ForAll and Exists quantifiers. Resolve[expr, dom]works over the domain dom. Common choices of dom are Complexes, Reals and Booleans.
• expr can contain equations, inequalities, domain specifications and quantifiers, in the same form as in Reduce.
• The result of Resolve[expr] always describes exactly the same mathematical set as expr, but without quantifiers.
• Resolve[expr] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
• When a quantifier such as ForAll[x, ...] is eliminated the result will contain no mention of the localized variable x.
• Resolve[expr] can in principle always eliminate quantifiers if expr contains only polynomial equations and inequalities over the reals or complexes.
• Resolve[expr] can in principles always eliminate quantifiers for any Boolean expression expr.
Prove that the unit disk is nonempty:
Find the conditions for a quadratic form over the reals to be positive:
Find conditions for a quadratic to have at least two distinct complex roots:
Prove that the unit disk is nonempty:
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Find the conditions for a quadratic form over the reals to be positive:
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Find conditions for a quadratic to have at least two distinct complex roots:
 Out[3]=
 Scope   (39)
Decide the existence of solutions of a univariate polynomial equation:
Decide the existence of solutions of a multivariate polynomial system:
Decide the truth value of fully quantified polynomial formulas:
Find conditions under which a polynomial equation has solutions:
Find conditions under which a polynomial system has solutions:
Find conditions under which a quantified polynomial formula is true:
Decide the existence of solutions of a univariate polynomial equation:
Decide the existence of solutions of a univariate polynomial inequality:
Decide the existence of solutions of a multivariate polynomial system:
Decide the truth value of fully quantified polynomial formulas:
Decide the existence of solutions of an exp-log equation:
Decide the existence of solutions of an elementary function equation in a bounded interval:
Decide the existence of solutions of a holomorphic function equation in a bounded interval:
Decide the existence of solutions of a periodic elementary function equation:
Fully quantified formulas exp-log in the first variable and polynomial in the other variables:
Fully quantified formulas elementary and bounded in the first variable:
Fully quantified formulas holomorphic and bounded in the first variable:
Find conditions under which a linear system has solutions:
Find conditions under which a quadratic system has solutions:
Find conditions under which a polynomial system has solutions:
Find conditions under which a formula linear in quantified variables is true:
Find conditions under which a formula quadratic in quantified variables is true:
Find conditions under which a quantified polynomial formula is true:
Decide the existence of solutions of a linear system of equations:
Decide the existence of solutions of a linear system of equations and inequalities:
Decide the existence of solutions of a univariate polynomial equation:
Decide the existence of solutions of a univariate polynomial inequality:
Decide the existence of solutions of Frobenius equations:
Decide the existence of solutions of binary quadratic equations:
Decide the existence of solutions of a Thue equation:
Decide the existence of solutions of a sum of squares equation:
Decide the existence of solutions of a bounded system of equations and inequalities:
Decide the existence of solutions of a system of congruences:
Decide the satisfiability of a Boolean formula:
Find conditions under which a quantified Boolean formula is true:
Decide the existence of solutions of an equation involving a real and a complex variable:
Decide the existence of solutions of an inequality involving Abs[x]:
Find under what conditions a fourth power of a complex number is real:
 Options   (4)
Here the solutions for x are expressed in terms of y:
With Backsubstitution->True, Resolve gives explicit numeric values for x:
By default Resolve does not use general formulas for solving cubics in radicals:
With , Resolve expresses roots of cubics in terms of radicals:
By default Resolve does not use general formulas for solving quartics in radicals:
With , Resolve expresses roots of quartics in terms of radicals:
This computation takes a long time due to high degrees of algebraic numbers involved:
With WorkingPrecision->100 we get an answer faster, but it may be incorrect:
 Applications   (6)
Find conditions for a quintic to have all roots equal:
Prove the inequality between the arithmetic mean and the geometric mean:
Prove a special case of Hölder's inequality:
Prove a special case of Minkowski's inequality:
Find conditions for a quadratic to be always positive:
Test whether one region is included in another:
Plot the relationship:
For fully quantified systems of equations and inequalities Resolve is equivalent to Reduce:
A solution instance can be found with FindInstance:
For systems with free variables Resolve may return an unsolved system:
Reduce eliminates quantifiers and solves the resulting system:
Eliminate can be used to eliminate variables from systems of complex polynomial equations:
Resolve gives the same equations, but may also give inequations:
Because x appears in an inequality, it is assumed to be real:
This allows complex values of x for which both sides of the inequality are real:
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