(mathematics) (d²ʸ)/(dx²)-xy=0,,!, the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential).

(mathematics) A polynomial all of whose coefficients are 1.

(mathematics, of a function) Being able to be locally represented by convergent power series around every point of the domain.

(algebra) A multilinear map, given by [x, y, z] = xy(z) − x(yz), that measures the degree of nonassociativity of a ring or algebra.

(mathematics) A spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of basis splines of that degree.

(combinatorics) Any member of a certain class of polynomials used in the study of set partitions, and related to Stirling numbers and Bell numbers.

(mathematics) A theorem stating that any non-singular algebraic curve defined by algebraic number coefficients represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only.

(combinatorics) A Boolean function f: Z ₂ⁿ→ Z ₂ whose Walsh transform has constant absolute value.

(mathematics) Any member of a certain group of polynomials remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up.

(mathematics) Pertaining to a permutation such that there is at most one left and one right descent (at most one index i and one index j, such that wsᵢ < w and sⱼw < w).

(mathematics) Either of two elements of a permutation group that trade places with each other.

(mathematics) A theorem that gives a lower bound on the size of a disc in which an inverse to a holomorphic function exists.

(mathematics) Any member of the sequence of polynomials Φₙ⁽ʳ⁾(x) given by generating functions of the form displaystyle C(ztʳB(t))=∑_(n>0)𝛷ₙ⁽ʳ⁾(z)tⁿ.

(mathematical analysis, of a function) Such that the inverse image of any open set in its codomain is a Borel set of its domain.

(mathematics) A generalization of this in Nachbin's theorem.

(mathematics) A polynomial in the coefficients of a form of even degree that vanishes when the form is a sum of an unusually small number of powers of linear forms.

(linear algebra, mathematical analysis) A theorem which states that the absolute value of the dot product between two vectors is less than or equal to the product of the magnitudes of the two vectors.

(mathematics) A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t).

(mathematics, of a set) Such that its image under the specified operation is contained in it.

(mathematics, of an operator) Having the property that there exists a constant L ≥ 0 such that for all x and y in the domain, is greater than or equal to 1/L ||Ax - Ay||², where Ax is the operator applied to x.

(mathematics, of a category) In which all small colimits exist.

(differential geometry) the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: (𝛼,𝛽)=∫_M𝛼∧ star 𝛽, so that (𝛼,d𝛽)=(𝛿𝛼,𝛽).

(mathematics) The general solution of a homogenous linear differential equation.

(mathematics, algebra) Whose coefficients are complex numbers; defined over the field of complex numbers.

(mathematics, complex analysis) The theorem that if a complex number is a root for a polynomial with real coefficients, the complex conjugate of that number will also be a root.

(algebra) More generally, any of a set of irrational or complex numbers that are zeros of the same polynomial with integral coefficients.

(mathematics) A spanning supergraph.

(physics) A differential operator which may be expressed as ∂_𝜇∂^𝜇=∑_(𝜇=0)³∂/∂x^𝜇∂/∂x_𝜇; it is the four-dimensional (Minkowski space) equivalent of the three-dimensional Laplace operator.

Any of a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support.

(mathematics) Any of several complex numbers that are coordinates of the vertices of a regular polygon of unit radius

(mathematics) A technique introduced by Alexander Grothendieck for comparing invariants about coherent sheaves on noetherian systems.

(algebra) An element of an algebra (the algebra of dual numbers) which includes the real numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.

(mathematics) A nonlinear second-order differential equation used to model certain damped and driven oscillators.

(algebra) a coefficient of some power of t in (t+x_1)(t+x_2)⋯(t+x_n), for some n∈ N

(complex analysis, of a complex function) Complex-differentiable on all of ℂ.

(mathematics) An integrable non-linear partial differential equation used in finding the exact solutions of Einstein's equations in the general theory of relativity.

(complex analysis) The equation e^(i𝜋)=-1,!, which unifies diverse fields of mathematics.

(mathematics) Injective in specific contexts, e.g. of representations in representation or functors in category theory.

(mathematics) In knot theory, a theorem stating that three-dimensional smooth curves with small total curvature must be unknotted.

(mathematics) Any member of a polynomial sequence which can be considered as a generalization of the Fibonacci numbers.

(algebra) A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas). Starting with a set of n variables which are independent of each other and are called generators, the power set of this set has 2ⁿmembers which may be called atoms and are valuations of the n variables: a valuation can be considered to be a set of variables which are "true" under that valuation, or a conjunction of generators (such that variables not included in that set are included in negated form in the equivalent conjunction). Then the power set of the set of atoms yields a set of 2^(2ⁿ) members which are the elements of the said field of sets. These elements correspond to Boolean formulas: a formula can be considered to be a set of valuations which make the formula true, or a linear combination (i.e., a disjunction) of atoms.

(mathematics, linear algebra, of a matrix) Having the highest possible rank for its size; having a rank (dimension) equal to the the number of columns or the number of rows.

(algebra) Any complex number of the form a + bi, where a and b are integers.

(mathematics) An algorithm for finding sums of hypergeometric terms that are themselves hypergeometric terms.

(mathematics) In the field of additive combinatorics, a class of norms on functions on a finite group or group-like object which quantify the amount of structure or randomness present.

(mathematics) A theorem stating that there is no nonvanishing continuous tangent vector field on the sphere; colloquially, that "you can't comb a hairy ball flat without creating a cowlick".

(mathematics) An integral transform closely related to the Fourier transform, but which transforms real-valued functions to real-valued functions.

(mathematics) A method of interpolating data points as a polynomial function.

(mathematics) Any member of a classical orthogonal polynomial sequence having diverse applications.

(geometry) a theorem stating that if a set of points P_i lying on a line is isometrically mapped to points P_i' on a different line, then the midpoints of the segments P_iP_i' also lie on a line.

(mathematics) homogeneous polynomial

(mathematics) Alternative spelling of homogeneous polynomial [(mathematics) a polynomial such that the sum of the exponents of the variables is the same for every term.]

(mathematics, computing theory) The rule stating that a polynomial of degree n can be (optimally) evaluated with only n multiplications and n additions: a_0+a_1x+a_2x²+a_3x³+⋯+a_nxⁿ\=a_0+x bigg (a_1+x Big (a_2+x big (a_3+⋯+x(a_n-1+x,a_n)⋯ big ) Big ) bigg ).

(mathematics) Pertaining to the continuation of a lower central series to infinite ordinal numbers via transfinite recursion.

(mathematics) The property of a number shared with another number with which it is hyperarithmetical.

(mathematics) Being or pertaining to a positive hyperinteger.

(mathematics) (said of an element of an algebraic structure with a binary operation, such as a group or semigroup) Such that, when it operates on itself, the result is equal to itself.

(algebraic geometry) The Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L.

(complex analysis, broad sense) A number of the form a + bi, where a and b are real numbers and b is nonzero.

(mathematics, countable) An injective function.

(mathematics) Clipping of irreducible representation.

(mathematics) Any member of a certain class of orthogonal polynomials.

(number theory) A mathematical function of integer a and odd positive integer b, generally written (a/b), based on, for each of the prime factors pᵢ of b, whether a is a quadratic residue or nonresidue modulo pᵢ.

(calculus) The determinant of such a matrix.

(mathematics) A famous problem on polynomials in several variables.

(mathematics) A particular knot polynomial that is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t^(1/2) with integer coefficients.

(quantum mechanics) A no-go theorem that places certain constraints on the permissible types of hidden-variable theories that try to explain the apparent randomness of quantum mechanics as a deterministic model featuring hidden states. It demonstrates the impossibility of quantum-mechanical observables representing "elements of physical reality".

(mathematics) For a given set of points (x_j,y_j) with no two x_j values equal, the polynomial of lowest degree that assumes at each value x_j the corresponding value y_j, so that the functions coincide at each point.

(mathematics) A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n.

(mathematics) A polynomial that is a solution to Laguerre's equation.

(differential geometry) A generalization of the Laplace operator; the anticommutator of the exterior derivative and the codifferential.

(mathematics) A linear combination of positive and negative powers of a variable with coefficients in a given field. They differ from ordinary polynomials in that they may have terms of negative degree.

(mathematics) Any of a certain system of complete and orthogonal polynomials, with numerous mathematical properties and applications.

(mathematics) The equation that defines the Legendre polynomials: d/dx[(1-x²)d/dxP_n(x)]+n(n+1)P_n(x)=0.

(mathematics, functional analysis) An operator L such that for functions f and g and scalar λ, L (f + g) = L f + L g and L λf = λ L f.

(algebra, of a set of vectors or ring elements) Having a nontrivial linear combination which is zero.

In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

(mathematics) Any member of a polynomial sequence which can be considered as a generalization of the Lucas numbers.

(mathematics) Any of a two-parameter family of orthogonal polynomials indexed by a positive weight of a root system.

(mathematics) A result about the decomposability of measure spaces, playing an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval [0,1] on the reals), and "purely atomic parts", using the counting measure on some discrete space.

(mathematical analysis) A measurable space which has a positive measure defined on its σ-algebra.

(complex analysis, of a function) That is the ratio of two holomorphic functions (and so possibly infinite at a discrete set of points).

(mathematics) A polynomial of the form displaystyle (1+t)ᶻ(1-t)⁻ᶻ=∑ₙg_n(z)tⁿ.

(complex analysis) A set in the complex plane obtained by attempting to extend a given complex-analytic function along rays emanating from a given point.

(mathematics) A branch of pure mathematics relating the Monster group to an invariant of elliptic functions.

(algebra) An element of an algebraic structure, generally denoted 1, which is an identity for a multiplicative operation (generally denoted × or *, or by concatenation).

(algebraic geometry) Of a line bundle on a complete algebraic variety over a field: such that the degree of its restriction to every algebraic curve in the variety is non-negative.

(mathematics, number theory) A cusp form that is "new" at a given level N, where the levels are the nested subgroups Γ₀(N) of the modular group, with N ordered by divisibility.

Alternative letter-case form of noetherian [(algebra) Of a ring in which any ascending chain of ideals eventually starts repeating.]

(algebra) A module whose every submodule is finitely generated; or equivalently, a module whose submodules satisfy an ascending chain condition (i.e., any ascending chain of submodules levels off after a finite number of steps).

(mathematics) a nonreal number

(𝜆𝜎₂(𝜆)+(log𝜆)𝜎₁(𝜆)) where a 𝜎₂ is holomorphic and nonvanishing at 0.

(mathematics, of a function) Injective, being an injection: having the property that no two elements of the domain are mapped to the same image.

(algebra) The sum of the exponents on the variables in a monomial, or the highest such among all monomials in a polynomial.

(mathematics) The determinant of a skew-symmetric matrix, capable of being written as the square of a polynomial in the matrix entries.

(taxonomy) of a polynomial name or entity

(algebra) Any algebraic equation in which one or both sides are in the form of a polynomial.

(algebra) A linear combination of powers of an indeterminate (or products of powers of more than one indeterminate), with coefficients belonging to an integral domain or a field. (The indeterminate is thought of as an element extraneous to the set of coefficients, instead of as a variable element of it (as in the case of polynomial functions), just as, say, the square root of negative one is an element extraneous to the set of integers when it is adjoined to them to form the domain of Gaussian integers. The indeterminate forms a free commutative monoid, to which all powers of it belong, and the unity of it can also show up implicitly in the constant term of a polynomial form.)

(mathematics, of a C*-algebra) In which 0 and 1 are the only projections.

(mathematics) Of a field: both quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over the field has a point defined over the field).

(mathematics) A set for which the number of elements that are not in the intersection (of two infinite sets) is finite.

(mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.

(linear algebra) A theorem about linear transformations (or the matrices that represent them) stating that the rank plus the nullity equals the dimension of the entire vector space (which is the linear transformation’s domain).

(mathematics, of a number) Being either a rational number, or the limit of a convergent infinite sequence of rational numbers: being one of a set of numbers with a one-to-one correspondence to the points on a line.

(complex analysis) Of a complex number a + bi, the value a.

(linear algebra) The form of a row-reduced matrix which is in row echelon form.

(mathematics) A mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules, and having applications in fluid dynamics and invariant theory.

(mathematics) A generalization of a Fourier series extended to situations in which a displacement field has an arbitrary time dependence while still maintaining its spatial periodicity.

(mathematics) A holomorphic function in the Schur class.

(singular only) A particular operator that, when applied to a function f yields a function x↦(f(x))/(f'(x))-3/2((f(x))/(f'(x)))²

(algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial).

(mathematical analysis) The property of a function f between metric spaces, that given a convergent sequence x_n→x^*, then f(x_n)→f(x^*), i.e. the property of a function that it preserves sequential convergence.

(mathematics) Of a group: having no normal subgroup.

(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0).

(mathematics) For some univariate polynomial p, a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials.

(mathematics)

(mathematics) surreal number

(mathematics) A matrix equation, used in control theory, of the form AX + XB = C. Given matrices A, B, and C, the problem is to find possible matrices for X.

(set theory) The set that contains all elements belonging to either of given two sets but not to the both; the relative complement of the intersection in the union of given two sets.

Alternative form of Taylor polynomial [(mathematical analysis) A truncated Taylor series; the sum of the first n terms of a Taylor series.]

(mathematics) Any member of a polynomial sequence of binomial type defined by T_n(x)=∑ₖ₌₀ⁿS(n,k)xᵏ=∑ₖ₌₀ⁿn atop kxᵏ, where S(n,k)=n atop kis a Stirling number of the second kind, i.e. the number of partitions of a set of size n into k disjoint non-empty subsets.

(adjective, linear algebra) The resulting matrix, derived from performing a transpose operation on a given matrix.

(mathematics, rare) A power series in several variables P:Kᵈ→K (with K a field) of the form P(x)=𝛴_(𝛼∈ℕᵈ)c_𝛼x^𝛼 whose coefficients are bounded in some specific sense.

(mathematics) A unitary matrix or operator.

(mathematics) Any of various formulas that relate the coefficients of a polynomial to a sum or product of its roots.