Resolve
✖
Resolve

Details and Options

- Resolve is in effect automatically applied by Reduce.
- expr can contain equations, inequalities, domain specifications, and quantifiers, in the same form as in Reduce.
- The statement expr can be any logical combination of:
-
lhs==rhs equations lhs!=rhs inequations lhs>rhs or lhs>=rhs inequalities expr∈dom domain specifications {x,y,…}∈reg region specification ForAll[x,cond,expr] universal quantifiers Exists[x,cond,expr] existential quantifiers - 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 principle always eliminate quantifiers for any Boolean expression expr.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Prove that the unit disk is nonempty:

https://wolfram.com/xid/0bn6rwi-e072jt

Find the conditions for a quadratic form over the reals to be positive:

https://wolfram.com/xid/0bn6rwi-jnjj6b

Find conditions for a quadratic to have at least two distinct complex roots:

https://wolfram.com/xid/0bn6rwi-fl732j

Find the projection of a geometric region:

https://wolfram.com/xid/0bn6rwi-3qq5us


https://wolfram.com/xid/0bn6rwi-8wyqkl

Scope (52)Survey of the scope of standard use cases
Complex Domain (6)
Decide the existence of solutions of a univariate polynomial equation:

https://wolfram.com/xid/0bn6rwi-o97tp

Decide the existence of solutions of a multivariate polynomial system:

https://wolfram.com/xid/0bn6rwi-ftau4n

Decide the truth value of fully quantified polynomial formulas:

https://wolfram.com/xid/0bn6rwi-g7hwsd


https://wolfram.com/xid/0bn6rwi-bmpaxf

Find conditions under which a polynomial equation has solutions:

https://wolfram.com/xid/0bn6rwi-drvio4

Find conditions under which a polynomial system has solutions:

https://wolfram.com/xid/0bn6rwi-nk5j0r

Find conditions under which a quantified polynomial formula is true:

https://wolfram.com/xid/0bn6rwi-byt21l

Real Domain (18)
Decide the existence of solutions of a univariate polynomial equation:

https://wolfram.com/xid/0bn6rwi-gyjz0

Decide the existence of solutions of a univariate polynomial inequality:

https://wolfram.com/xid/0bn6rwi-fwu95i

Decide the existence of solutions of a multivariate polynomial system:

https://wolfram.com/xid/0bn6rwi-gr7crv

Decide the truth value of fully quantified polynomial formulas:

https://wolfram.com/xid/0bn6rwi-brp8ku


https://wolfram.com/xid/0bn6rwi-cpxl1h

Decide the existence of solutions of an exp-log equation:

https://wolfram.com/xid/0bn6rwi-gvp15g

Decide the existence of solutions of an exp-log inequality:

https://wolfram.com/xid/0bn6rwi-kyie2

Decide the existence of solutions of an elementary function equation in a bounded interval:

https://wolfram.com/xid/0bn6rwi-imjan8

Decide the existence of solutions of a holomorphic function equation in a bounded interval:

https://wolfram.com/xid/0bn6rwi-nl029


Decide the existence of solutions of a periodic elementary function equation:

https://wolfram.com/xid/0bn6rwi-dm2lpf

Fully quantified formulas exp-log in the first variable and polynomial in the other variables:

https://wolfram.com/xid/0bn6rwi-e6l4v8


https://wolfram.com/xid/0bn6rwi-b5zzk3

Fully quantified formulas elementary and bounded in the first variable:

https://wolfram.com/xid/0bn6rwi-8arqe


https://wolfram.com/xid/0bn6rwi-ec9nd2

Fully quantified formulas holomorphic and bounded in the first variable:

https://wolfram.com/xid/0bn6rwi-epcw0p


https://wolfram.com/xid/0bn6rwi-bkdxj

Find conditions under which a linear system has solutions:

https://wolfram.com/xid/0bn6rwi-m9jkw

Find conditions under which a quadratic system has solutions:

https://wolfram.com/xid/0bn6rwi-e17391

Find conditions under which a polynomial system has solutions:

https://wolfram.com/xid/0bn6rwi-dijh5x

Find conditions under which a formula linear in quantified variables is true:

https://wolfram.com/xid/0bn6rwi-lfcchk

Find conditions under which a formula quadratic in quantified variables is true:

https://wolfram.com/xid/0bn6rwi-c129o7

Find conditions under which a quantified polynomial formula is true:

https://wolfram.com/xid/0bn6rwi-kdrucq

Integer Domain (10)
Decide the existence of solutions of a linear system of equations:

https://wolfram.com/xid/0bn6rwi-sz2ju

Decide the existence of solutions of a linear system of equations and inequalities:

https://wolfram.com/xid/0bn6rwi-fulreq

Decide the existence of solutions of a univariate polynomial equation:

https://wolfram.com/xid/0bn6rwi-km7sh0

Decide the existence of solutions of a univariate polynomial inequality:

https://wolfram.com/xid/0bn6rwi-36h74

Decide the existence of solutions of Frobenius equations:

https://wolfram.com/xid/0bn6rwi-cjs3ts


https://wolfram.com/xid/0bn6rwi-deaki

Decide the existence of solutions of binary quadratic equations:

https://wolfram.com/xid/0bn6rwi-g1rl8c


https://wolfram.com/xid/0bn6rwi-bk2dj1

Decide the existence of solutions of a Thue equation:

https://wolfram.com/xid/0bn6rwi-enjls9

Decide the existence of solutions of a sum of squares equation:

https://wolfram.com/xid/0bn6rwi-hsrd0b

Decide the existence of solutions of a bounded system of equations and inequalities:

https://wolfram.com/xid/0bn6rwi-jqvrli

Decide the existence of solutions of a system of congruences:

https://wolfram.com/xid/0bn6rwi-cpi4z8

Boolean Domain (2)
Finite Field Domains (5)
Decide the existence of solutions of univariate equations:

https://wolfram.com/xid/0bn6rwi-cfndqf


https://wolfram.com/xid/0bn6rwi-gyftoc

Verify that a univariate equation is satisfied by all field elements:

https://wolfram.com/xid/0bn6rwi-f6gotk

Decide the existence of solutions of systems of linear equations:

https://wolfram.com/xid/0bn6rwi-hef9nj


https://wolfram.com/xid/0bn6rwi-dyppb

Decide the existence of solutions of systems of polynomial equations:

https://wolfram.com/xid/0bn6rwi-fi2b3p


https://wolfram.com/xid/0bn6rwi-fna1o8


https://wolfram.com/xid/0bn6rwi-c3cw8z


https://wolfram.com/xid/0bn6rwi-ixizko

Mixed Domains (3)
Decide the existence of solutions of an equation involving a real and a complex variable:

https://wolfram.com/xid/0bn6rwi-gni56x

Decide the existence of solutions of an inequality involving Abs[x]:

https://wolfram.com/xid/0bn6rwi-gle2zl

Find under what conditions a fourth power of a complex number is real:

https://wolfram.com/xid/0bn6rwi-fhjebh

Geometric Regions (8)

https://wolfram.com/xid/0bn6rwi-fqngja

https://wolfram.com/xid/0bn6rwi-eckgme


https://wolfram.com/xid/0bn6rwi-bwhcrz

https://wolfram.com/xid/0bn6rwi-d2bvqv

Project a cone to the -
plane:

https://wolfram.com/xid/0bn6rwi-b9wofb

https://wolfram.com/xid/0bn6rwi-j0un4z


https://wolfram.com/xid/0bn6rwi-tc2ldk


https://wolfram.com/xid/0bn6rwi-vvkhdz

https://wolfram.com/xid/0bn6rwi-b8ttk3

A parametrically defined region:

https://wolfram.com/xid/0bn6rwi-juwcyb

https://wolfram.com/xid/0bn6rwi-fo5so0


https://wolfram.com/xid/0bn6rwi-mi7sv6

https://wolfram.com/xid/0bn6rwi-25kgk


https://wolfram.com/xid/0bn6rwi-r1kcx

Regions dependent on parameters:

https://wolfram.com/xid/0bn6rwi-9d869w

https://wolfram.com/xid/0bn6rwi-fuuwc4

The conditions on indicate when the line intersects the circle:

https://wolfram.com/xid/0bn6rwi-ddvzpv


https://wolfram.com/xid/0bn6rwi-k1jnua

https://wolfram.com/xid/0bn6rwi-h8cff


https://wolfram.com/xid/0bn6rwi-cuvibo


https://wolfram.com/xid/0bn6rwi-c4gkcy

https://wolfram.com/xid/0bn6rwi-ft2kkm

Options (4)Common values & functionality for each option
Backsubstitution (1)
Cubics (1)
Quartics (1)
WorkingPrecision (1)
This computation takes a long time due to high degrees of algebraic numbers involved:

https://wolfram.com/xid/0bn6rwi-joy4r1

With WorkingPrecision->100 you get an answer faster, but it may be incorrect:

https://wolfram.com/xid/0bn6rwi-kjnccl

Applications (9)Sample problems that can be solved with this function
Polynomials (2)
Theorem Proving (3)
Prove the inequality between the arithmetic mean and the geometric mean:

https://wolfram.com/xid/0bn6rwi-d09mxl

Prove a special case of Hölder's inequality:

https://wolfram.com/xid/0bn6rwi-jc94i0

Prove a special case of Minkowski's inequality:

https://wolfram.com/xid/0bn6rwi-bc5cii

Geometry (4)
The region is a subset of
,
if
is true. Show that Disk[{0,0},{2,1}] is a subset of Rectangle[{-2,-1},{2,1}]:

https://wolfram.com/xid/0bn6rwi-18g7a

https://wolfram.com/xid/0bn6rwi-c2c6yf


https://wolfram.com/xid/0bn6rwi-dplxmk

Show that Cylinder[]⊆Ball[{0,0,0},2]:

https://wolfram.com/xid/0bn6rwi-bthjls

https://wolfram.com/xid/0bn6rwi-ba7xve


https://wolfram.com/xid/0bn6rwi-fn0f55

A region is disjoint from
if
. Show that Circle[{0,0},2] and Disk[{0,0},1] are disjoint:

https://wolfram.com/xid/0bn6rwi-iczok
There are no points in the intersection, so they are disjoint:

https://wolfram.com/xid/0bn6rwi-h6331u


https://wolfram.com/xid/0bn6rwi-ei5vty

A region intersects
if
. Show that Circle[{0,0},1] intersects Disk[{1/2,0},1]:

https://wolfram.com/xid/0bn6rwi-j5q9j
There are points in the intersection:

https://wolfram.com/xid/0bn6rwi-ekg231


https://wolfram.com/xid/0bn6rwi-rvbro

Properties & Relations (5)Properties of the function, and connections to other functions
For fully quantified systems of equations and inequalities, Resolve is equivalent to Reduce:

https://wolfram.com/xid/0bn6rwi-nl87yt


https://wolfram.com/xid/0bn6rwi-f9nyzh

A solution instance can be found with FindInstance:

https://wolfram.com/xid/0bn6rwi-bb52em

For systems with free variables, Resolve may return an unsolved system:

https://wolfram.com/xid/0bn6rwi-et9t7m

Reduce eliminates quantifiers and solves the resulting system:

https://wolfram.com/xid/0bn6rwi-dfsd54

Eliminate can be used to eliminate variables from systems of complex polynomial equations:

https://wolfram.com/xid/0bn6rwi-gqu0wc

Resolve gives the same equations, but may also give inequations:

https://wolfram.com/xid/0bn6rwi-jni7vp

Find a formula description of a TransformedRegion:

https://wolfram.com/xid/0bn6rwi-mxt4j


https://wolfram.com/xid/0bn6rwi-gwbfi2

Compute a formula description of using RegionMember:

https://wolfram.com/xid/0bn6rwi-ddt44r

Check that the formulas are equivalent:

https://wolfram.com/xid/0bn6rwi-mou1qy

Resolve shows that the polynomial is non-negative:

https://wolfram.com/xid/0bn6rwi-b2qs46

https://wolfram.com/xid/0bn6rwi-cqgp6y

Use PolynomialSumOfSquaresList to represent as a sum of squares:

https://wolfram.com/xid/0bn6rwi-gucy7i


https://wolfram.com/xid/0bn6rwi-bmrxo2

The Motzkin polynomial is non-negative, but is not a sum of squares:

https://wolfram.com/xid/0bn6rwi-ltua3e

https://wolfram.com/xid/0bn6rwi-qm557


https://wolfram.com/xid/0bn6rwi-2zhfo


Possible Issues (1)Common pitfalls and unexpected behavior
Wolfram Research (2003), Resolve, Wolfram Language function, https://reference.wolfram.com/language/ref/Resolve.html (updated 2024).
Text
Wolfram Research (2003), Resolve, Wolfram Language function, https://reference.wolfram.com/language/ref/Resolve.html (updated 2024).
Wolfram Research (2003), Resolve, Wolfram Language function, https://reference.wolfram.com/language/ref/Resolve.html (updated 2024).
CMS
Wolfram Language. 2003. "Resolve." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Resolve.html.
Wolfram Language. 2003. "Resolve." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Resolve.html.
APA
Wolfram Language. (2003). Resolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Resolve.html
Wolfram Language. (2003). Resolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Resolve.html
BibTeX
@misc{reference.wolfram_2025_resolve, author="Wolfram Research", title="{Resolve}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Resolve.html}", note=[Accessed: 19-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_resolve, organization={Wolfram Research}, title={Resolve}, year={2024}, url={https://reference.wolfram.com/language/ref/Resolve.html}, note=[Accessed: 19-June-2025
]}