For nonnegative integers $a_1,\dots ,a_n$,

If polynomials $f_1,\dots ,f_r$ over a finite field have total degree sum strictly smaller than their number of variables, then the number of their common zeros is divisible by the field characteristic.

The Karp–Lipton theorem states that $\mathbf{NP}\subseteq \mathbf{P}/\mathrm{poly}$ implies that the polynomial hierarchy collapses to its second level.

A language is in ${\Sigma }_2^{\mathbf{P}}$ when membership has the form $\exists u\forall vR(x,u,v)$ for a polynomial-time predicate $R$ and polynomially bounded strings. Reversing the quantifiers defines ${\Pi }_2^{\mathbf{P}}$.

A polynomial equation has the form $p(x_1,\dots ,x_n)=0$ for a polynomial $p$. Its solutions are the points at which that polynomial vanishes.

The magic state $|A_{\theta }⟩=(|0⟩+e^{i\theta }|1⟩)/\sqrt{2}$ and a short controlled-NOT gate circuit can apply either $P(\theta )$ or $P(-\theta )$ to an input, with a measured bit identifying the branch.

The controlled phase gate contributes $e^{i\varphi }$ exactly when both of its computational-basis input bits are one. It is diagonal and symmetric between its two qubits.

The equivalence theorem states that an invertible local field redefinition leaves on-shell S-matrix elements unchanged, although it changes off-shell correlation functions and individual interaction vertices.

The LSZ reduction formula obtains an S-matrix element from a time-ordered correlation function by amputating external propagators, taking external momenta on shell, and multiplying by pole-residue factors and external wavefunctions.

A generating functional is a path integral with a source $J$ coupled to a field. Its functional derivatives with respect to $J$ generate time-ordered correlation functions.

A Wick rotation deforms a time or energy contour from real to imaginary values, converting suitable Minkowski integrals into Euclidean integrals while respecting pole prescriptions.

The Helmholtz equation is the constant-coefficient elliptic equationOn a bounded domain, nonzero Dirichlet solutions occur when $k^2$ is an eigenvalue of the negative Laplacian.

The elliptic Liouville equation $\Delta u=e^{2u}$ describes conformal metrics of constant negative Gaussian curvature under a common sign convention.

A relativized complexity class ${\mathcal{C}}^A$ allows every machine defining $\mathcal{C}$ to query the same oracle $A$.

$\mathbf{co}\text{-}\mathbf{RP}$ consists of complements of languages in RP. Its algorithms always accept positive instances and reject negative instances with probability at least one half.

$\mathbf{RP}$ contains decision problems with a polynomial-time randomized algorithm that rejects every negative instance and accepts every positive instance with probability at least one half.

Gaussian null coordinates adapt one coordinate to the generators of a null hypersurface, one affine coordinate to transverse null geodesics, and the remaining coordinates to a spacelike cross-section. If the generator parameter is affine, the metric can locally be written

A Killing horizon is a null hypersurface whose null generators are orbits of a Killing vector field $K$. On the horizon, $K^b{\nabla }_b{K}^a=\kappa K^a$ defines its surface gravity.

An $R$-module is Noetherian when every submodule is finitely generated, equivalently when every ascending chain of submodules stabilizes. Quotients and finite direct sums of Noetherian modules are Noetherian.

An automorphism is an isomorphism $f:X\to X$: a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible endomorphism. Its inverse satisfieswhere $1_X$ is the identity morphism. Both maps must be morphisms in the chosen category. For ordinary groups, rings, and vector spaces over a fixed field, a bijective morphism automatically has a compatible inverse. For topological spaces, an automorphism is a homeomorphism from the space to itself: a continuous bijection needs a continuous inverse as well. See Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11.

For example, $T(x,y)=(x+y,y)$ is an automorphism of the vector space ${\mathbb{R}}^2$ over the real numbers, with inverse $T^{-1}(u,v)=(u-v,v)$. Both are linear maps. The linear endomorphism $P(x,y)=(x,0)$ from endomorphism preserves addition and scalar multiplication too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.

The automorphisms of a fixed object in a locally small category form a group under composition in a category: the identity morphism is the identity element, and the inverse element of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a group, the resulting automorphism group records which relabelings preserve all products; the examples in group automorphism show how restrictive that condition can be.