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If the Hilbert polynomial of the homogeneous coordinate ring of a projective curve isthen $\mathrm{deg}C=d$. Equivalently, a sufficiently general hyperplane section has scheme-theoretic length $d$.

The graph $\{(x,e^x):x\in \mathbb{C}\}$ is Zariski dense in ${\mathbb{A}}_{\mathbb{C}}^2$. If ${\sum }_j{p}_j(x)e^{jx}=0$, repeated application of $D-m$ removes the largest exponential term while acting injectively on the others, proving inductively that every $p_j$ vanishes.

For an algebraically closed field $k$ and an ideal $I\subseteq k[x_1,\dots ,x_n]$, the strong Hilbert Nullstellensatz isIts weak form says that an ideal with empty affine zero set is the unit ideal. Equivalently, if finitely many polynomials have no common zero, a polynomial combination of them equals one.

Every finitely generated algebra $A$ over a field has algebraically independent elements $y_1,\dots ,y_d$ such that $A$ is finite as a module over the polynomial subalgebra $k[y_1,\dots ,y_d]$. The integer $d$ is the Krull dimension of $A$.

A Zariski-open set is the complement of a Zariski-closed set. A distinguished affine open has the form $D(f)=\{x:f(x)\ne 0\}$.

A Zariski-closed set is a common zero set of polynomials. Arbitrary intersections and finite unions of such sets are again Zariski closed.

A nonempty topological space is irreducible when it is not the union of two proper closed subsets. Equivalently, every two nonempty open subsets intersect.

An algebraic variety over a field $k$ is, in a common modern convention, a reduced scheme that is a separated scheme of finite type over $k$. Some authors also require it to be irreducible; this wiki states that extra condition when reducible spaces matter. A classical convention over an algebraically closed field describes the same geometry using polynomial zero sets and their regular functions.

An affine variety has a coordinate ring $A$, a finitely generated reduced $k$-algebra, and its scheme model is $\mathrm{Spec}A$. General algebraic varieties glue finitely many such affine charts along open subsets. The regular functions are part of the structure: their local rings record which functions can be evaluated near each point.

For example, $x^2+y^2=1$ defines a real algebraic curve, while $y^2=x^3$ defines a cuspidal cubic. Both are one-dimensional, but the cusp has a singular local ring at the origin. The scheme defined by $x^2=0$ additionally remembers a nonzero nilpotent element; it is not a reduced scheme and so is outside this variety convention. This illustrates why an underlying zero set alone cannot describe every scheme.

On an integral scheme $X$, a Cartier divisor is represented on an open cover $U_i$ by nonzero rational functions $f_i$, with $f_i/f_j$ a regular unit on every overlap. Changing the local equation by a regular unit leaves the divisor unchanged. A principal Cartier divisor uses one global nonzero rational function on all the charts.

The transition ratios glue the local rank-one modules $f_i^{-1}{\mathcal{O}}_{U_i}$ into the divisor line bundle ${\mathcal{O}}_X(D)$. If each $f_i$ is regular, it cuts out an effective Cartier divisor. Allowing poles gives signed divisor data.

On the projective line, the coordinate $t$ defines the principal Cartier divisor $[0]-[\infty ]$: it has one simple zero at zero and one simple pole at infinity. For a general scheme, the analogous definition uses invertible elements of the sheaf of total quotient rings, so nonzero elements must be distinguished from non-zero-divisors.^[ref]

A Weil divisor on a Noetherian integral scheme is a finite integer combination of integral codimension-one closed subschemes.

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space together with a sheaf of rings describing its local functions. For an open subset $U$, ${\mathcal{O}}_X(U)$ is the ring of sections over $U$, with compatible restrictions and gluing. A smooth manifold with its real-valued smooth functions is a ringed space; a complex manifold with holomorphic functions is another. The underlying topology alone does not distinguish these kinds of functions. A locally ringed space additionally requires every stalk to be a local ring, which is the condition used in the definition of a scheme.

An associative algebra is an algebra over a field whose multiplication satisfies $(ab)c=a(bc)$. Unless stated otherwise, the algebras considered here have a multiplicative identity.

A Hopf algebra over a field $k$ is a bialgebra $H$ with an antipode $S$ satisfying $m(S\otimes 1)\Delta =u\epsilon =m(1\otimes S)\Delta$, where $m,u,\Delta ,\epsilon$ denote multiplication, unit, coproduct, and counit. These identities generalize taking an inverse.

For a group algebra $k[G]$, define $\Delta (g)=g\otimes g$, $\epsilon (g)=1$, and $S(g)=g^{-1}$, then extend linearly. The coproduct makes a group element act simultaneously on both factors of a tensor product of representations. A Hopf algebra therefore packages not only multiplication but also how symmetries act on products and how they are reversed.

For a vector space $V$ over a field of characteristic different from two and a quadratic form $Q$, its Clifford algebra is the quotient of the tensor algebra by the relations $v^2=Q(v)1$. Writing $B(u,v)=(Q(u+v)-Q(u)-Q(v))/2$ gives $uv+vu=2B(u,v)1$.

For a real two-dimensional positive-definite form and an orthonormal basis $e_1,e_2$, the generators satisfy $e_1^2=e_2^2=1$ and $e_1{e}_2=-e_2{e}_1$. The algebra has basis $1,e_1,e_2,e_1{e}_2$. Its Clifford modules provide spinor spaces, while products of suitable unit vectors represent rotations. The sign convention on $Q$ must be stated, especially when comparing geometric and physical conventions.

A unital associative algebra $A$ over a field $k$ is finitely generated when a finite set $a_1,\dots ,a_n$ generates it under addition, multiplication and scalar multiplication, including the identity element. Equivalently, every element is a finite linear combination of words in the generators. The words retain their order if multiplication is noncommutative.

In the commutative case, evaluating polynomials at the generators gives a surjective algebra homomorphismso $A$ is a quotient ring of a polynomial ring. This equivalence assumes commutativity. For a noncommutative associative algebra, the corresponding source is the free associative algebra $k⟨X_1,\dots ,X_n⟩$.^[ref]

Finite generation as an algebra differs from being finite-dimensional as a vector space. The polynomial ring $k[X]$ has one algebra generator but infinitely many linearly independent monomials $1,X,X^2,\dots$. By contrast, the quotient ring $k[X]/(X^2)$ has basis $1,\bar{X}$, and its relation truncates every higher power.

For birth rate $b(a)$, death rate $d(a)$ and exponential population growth rate $r$, the Euler-Lotka equation is

The Hubble time $H_0^{-1}$ is the expansion timescale inferred from the present Hubble parameter.

A Freiman homomorphism preserves additive quadruples: whenever $x_1+x_2=x_3+x_4$ in its domain, its values satisfy $\varphi (x_1)+\varphi (x_2)=\varphi (x_3)+\varphi (x_4)$.

For a positive integer $k$ and a function $f:G\to \mathbb{C}$ on a finite abelian group, with uniform expectations, definewhere $\mathcal{C}$ is complex conjugation. The alternating conjugations measure agreement around additive cubes. The expression is nonnegative; for $k\ge 2$ its $2^k$th root is a norm. At $k=1$ it is only the seminorm $|\mathbb{E}f|$.

With the multiplicative derivative ${\partial }_{h}f(x)=f(x+h)\overline{f(x)}$,Repeated multiplicative derivatives remove low-degree polynomial phases: a unit-modulus phase of degree less than $k$ has $U^k$ norm one. The Gowers U2 norm is the fourth moment of Fourier coefficients on a finite abelian group and detects linear correlations; the Gowers U3 norm controls four-term arithmetic progression counts through the generalized von Neumann inequality for four-term progressions.^[ref]

An almost period of a function $F$ in a normed space is a shift $t$ for which $\Vert {\tau }_{t}F-F\Vert$ is small.