homogeneous

adjective

UShoh-moh-JEE-nyuhs/ˌhoʊ.moʊˈd͡ʒiː.njəs/hom-uh-JEE-nee-uhs/ˌhɒm.ə(ʊ)ˈd͡ʒiː.nɪəs/

12 meanings of homogeneous

1
same throughoutadjective
Consistent in composition throughout, so that every part is made up the same way.
Blend the ingredients until the dough is homogeneous and no dry flour remains.
Under the microscope the sample proved homogeneous, with no visible layers or grains.
Similaruniform, consistent
Oppositeheterogeneous
2
same kindadjective
Of the same kind; alike or similar to one another.
The early settlers formed a homogeneous community, sharing one language and one church.
The committee was criticized as homogeneous, with every member drawn from the same industry.
Similaruniform
Oppositediverse, heterogeneous, mixed, varied
Note
The root meaning still shows: Greek homós, "same", and génos, "kind", reached English through Medieval Latin homogeneus. Said of people, it can describe unity or criticize sameness (a community sharing one language and church, or a committee drawn from one industry) and only the rest of the sentence tells which.
Checked 31 Oct 2026 · 3 facts
The Greek root homós means “same”. en.wiktionary.org
The Greek root génos means “kind”. en.wiktionary.org
Homogeneous reached English through Medieval Latin homogeneus, derived from Greek. en.wiktionary.org
3
chemistryadjective
In chemistry, existing in or forming a single phase, as when a catalyst is dissolved in the same liquid or gas as the reactants.
The chemists chose a homogeneous catalyst that dissolves in the same solvent as the reactants.
Because both reactants are gases that mix freely, the reaction between them is homogeneous.
Oppositeheterogeneous
Note
In chemistry the word is about phases: a homogeneous reaction or catalyst works within a single phase, the catalyst dissolved in or mixed into the same gas or liquid as the reactants. Its counterpart is heterogeneous catalysis, where something like a solid metal in a stream of gas does its work at the boundary between phases.
Checked 31 Oct 2026 · 4 facts
Homogeneous reactions and homogeneous catalysis involve a single phase. en.wikipedia.org
A homogeneous catalyst shares the gas or liquid solution phase of the reactants. preparatorychemistry.com
Heterogeneous catalysis involves different phases and reaction at their boundary. openstax.org
A solid metal exposed to flowing gas can act as a heterogeneous catalyst. openstax.org
4
space (symmetry)adjective
Of a space with a group action, the same everywhere: the group acts transitively by automorphisms, carrying any point to any other while preserving the structure (some authors also require a faithful action).
The sphere is homogeneous: a rotation can carry any point on it to any other point.
Euclidean space is a homogeneous space under the group of rigid motions.
Note
Such a space has no privileged points, and its standard description shows why: fix one point, gather the symmetries that keep it in place into a subgroup, and the whole space can be rebuilt as the group divided by that subgroup. The sphere arises this way from the rotations of three-dimensional space, with the rotations about one axis as the subgroup.
Checked 31 Oct 2026 · 3 facts
A homogeneous space has no privileged points: its symmetry group acts transitively. math.mit.edu
The symmetries fixing a chosen point form its stabilizer subgroup, and the homogeneous space is identifiable with the coset space of the group by that subgroup. ocw.mit.edu
The ordinary sphere is SO(3)/SO(2), with SO(2) realized as rotations about a fixed axis. physics.rutgers.edu
5
adjective
Of a polynomial, having every nonzero term of the same degree; more generally, of an element of a graded ring, lying wholly in one graded component.
The polynomial x³ + 2xy² − y³ is homogeneous of degree 3.
The polynomial x² + 3xy + y² is homogeneous because every term has degree two.
Note
The degree travels with the adjective: a polynomial is homogeneous of degree 2, of degree 3, and so on. Algebraists also call such polynomials forms, so a homogeneous polynomial of degree 2 is a quadratic form and one of degree 3 a cubic form.
Checked 31 Oct 2026 · 4 facts
Homogeneous polynomials have a specified degree, including degrees 2 and 3. math.uchicago.edu
Homogeneous polynomials are also called forms. math.uchicago.edu
A homogeneous polynomial of degree 2 is a quadratic form. math.uwaterloo.ca
A homogeneous polynomial of degree 3 is a cubic form. d-nb.info

Less common meanings

6
adjective
Of a function (or, by extension, a distribution), such that scaling all its inputs by one factor multiplies the output by a fixed power of that factor, called the degree.
The function f(x, y) = x² + y² is homogeneous of degree 2, since f(tx, ty) = t²f(x, y).
A production function that is homogeneous of degree 1 shows constant returns to scale.
Note
Analysis generalizes the polynomial meaning to any function that scales cleanly, and the degree need not be positive: 1/x is homogeneous of degree −1. Differentiable functions of this kind obey a tidy identity: multiply each input by the rate at which the output changes with it, add the results, and the total is the function times its degree.
Checked 31 Oct 2026 · 3 facts
Homogeneous functions generalize the scaling property of homogeneous polynomials. web.mit.edu
The degree need not be positive. Applying the cited scaling definition, 1/(λx) = λ⁻¹(1/x) for x ≠ 0 and λ > 0, confirming degree −1. courses.ems.psu.edu
A differentiable homogeneous function of degree k satisfies Euler’s identity: the sum of each coordinate times its corresponding partial derivative equals k times the function. mjo.osborne.economics.utoronto.ca
7
differential scalingadjective
Of a differential equation, homogeneous in the dependent variable and its derivatives: scaling them all by one factor scales the expression by a fixed power of that factor.
Since every term of y·y″ − 2(y′)² = 0 has degree two in y and its derivatives, the equation is homogeneous.
Because y·y″ + (y′)² = 0 is homogeneous, the substitution v = y′/y reduces it to a first-order equation.
Note
Here the scaling test of homogeneous functions is applied to y and its derivatives together. For a linear equation it comes to the same thing as the zero-free-term meaning, since every remaining term then has degree one; but it also reaches nonlinear equations such as y·y″ − 2(y′)² = 0, where every term has degree two.
Checked 31 Oct 2026 · 3 facts
Homogeneity in y and its derivatives applies the homogeneous-function scaling test to those quantities together. homepage.villanova.edu
For a linear differential equation, this homogeneity is equivalent to having no term independent of y and its derivatives; the remaining terms have degree one. macs.hw.ac.uk
The nonlinear equation y·y″ − 2(y′)² = 0 is homogeneous of degree two. Applying the source's criterion, both terms acquire the factor λ² when y and its derivatives are scaled by λ. sites.math.duke.edu
8
first-order equationadjective
Of a first-order differential equation, able to be written as f(x, y) dy = g(x, y) dx, where f and g are homogeneous functions of the same degree.
The equation dy/dx = (x + y)/x is homogeneous, so the substitution y = vx turns it into a separable equation.
Homogeneous equations like (x² + y²) dx = 2xy dy can be solved by setting y = vx.
Note
Here the test is a ratio: the equation can be rewritten so that y and x appear only as y/x, which is why the substitution y = vx makes it separable. The meanings can clash: dy/dx = (x + y)/x is homogeneous in this sense, yet under the linear-equation meaning it is not, since y′ − y/x = 1 keeps a term free of y.
Checked 31 Oct 2026 · 3 facts
The derivative can be expressed as a function of y/x alone. math.purdue.edu
The substitution y = vx transforms such an equation into a separable equation: v + xv′ = f(v), hence xv′ = f(v) − v. www-thphys.physics.ox.ac.uk
The example is homogeneous in the first-order ratio sense but nonhomogeneous as a linear equation: (x + y)/x = 1 + y/x, and its linear forcing term is 1. sites.math.duke.edu
9
adjective
In geometry, relating to homogeneous coordinates.
10
adjective
Of a linear equation, system of linear equations, or linear differential equation, having every term free of the unknowns equal to zero, so that the zero solution always satisfies it.
11
adjective
Of a relation, holding between a set and itself, so that it relates members of one set to members of the same set; being an endorelation.
12
adjective
Of a linear map between graded vector spaces, respecting the gradings: it sends the degree-j component into the degree-(i + j) component, for a fixed i, the map's degree.

Practice

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Origin

Greek homós ("same") and génos ("kind") combined in Ancient Greek homogenḗs, meaning "of the same kind, family or race," which Medieval Latin took over as homogeneus. The word can be compared with homo- ("same") and the adjective suffix -ous.