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An invertible linear map of the underlying vector space induces a projective linear transformation of ${\mathbb{P}}^n$; scalar multiples induce the same transformation.

The projective plane is the two-dimensional projective space ${\mathbb{P}}^2$.

A projective point is an equivalence class $[x_0:\cdots :x_n]$ of nonzero coordinate vectors under multiplication by a nonzero scalar.

For a field $k$,This nonvanishing obstructs affine covers of ${\mathbb{P}}_k^n$ having only $n$ members.

For $n>k\ge 0$, take an $(n-k)$-dimensional irreducible projective variety with one singular point and form its product with ${\mathbb{P}}^k$. The Segre embedding makes the product projective, and its singular locus is the singular point times ${\mathbb{P}}^k$, of dimension $k$. A quadric cone works in dimension at least two and a cuspidal cubic works in dimension one.

In the standard graded homogeneous coordinate ring $S=k[x_0,\dots ,x_n]$ of ${\mathbb{P}}^n$, the irrelevant ideal isIt vanishes only at the origin of the corresponding affine cone, which does not represent a projective point.

It is the polynomial-ring instance of the irrelevant ideal of a graded ring.

Lift the image of each basis element independently through the given surjection, then extend the chosen lifts linearly.

The naive height of a polynomial is the largest absolute value of its coefficients. For a primitive polynomial in $\mathbb{Z}[X_1,\dots ,X_n]$, this equals the projective height of its coefficient vector.

If the probability current of a normalizable wavefunction vanishes at spatial infinity, integrating the probability continuity equation over all space shows that the total probability $\int |\psi {|}^2$ is constant in time.

A hypersurface $\Sigma =\{\varphi =0\}$ is characteristic for a differential equation at $x\in \Sigma$ when its principal symbol vanishes on the conormal $d\varphi (x)$:It is non-characteristic where this quantity is nonzero.

A Euclidean domain is an integral domain $R$ with a function $\delta :R\setminus \{0\}\to \mathbb{N}$ such that, for $a,b\in R$ with $b\ne 0$, there are $q,r\in R$ satisfying $a=bq+r$ and either $r=0$ or $\delta (r)<\delta (b)$. The function need not be an ordinary absolute value.

The integers use $\delta (b)=|b|$ and $k[X]$ over a field uses polynomial degree. Repeated division gives the Euclidean algorithm and a greatest common divisor. Choosing a nonzero element of least Euclidean value in an ideal proves that every ideal is principal, so every Euclidean domain is a principal ideal domain.

Every submodule of a finite-rank free module over a principal ideal domain is free. Induction on rank splits off a generator of the image under one coordinate projection and applies the result to the kernel.

If distinct points $p,q$ on a smooth projective curve satisfy $(f)=p-q$, then $f:C\to {\mathbb{P}}^1$ is a finite morphism of degree one. Such a morphism between smooth projective curves is an isomorphism, so $C$ has genus zero. Hence no divisor $p-q$ on a curve of positive genus is principal.

Every degree-zero divisor on the projective line is principal. Indeed, if $D={\sum }_i{a}_i{p}_i$ and $L_i$ is a homogeneous linear form vanishing at $p_i$, then ${\sum }_i{a}_i=0$ makesa degree-zero rational function with $(f)=D$. Consequently, two divisors on ${\mathbb{P}}^1$ are linearly equivalent exactly when they have the same degree.

Over a UFD, products of primitive polynomials are primitive and factorization over the fraction field descends after content is removed.

The Möbius function stays far in pretentious distance from every Archimedean character. One useful uniform form isThis follows by estimating the prime sum ${\sum }_{p\le x}(1+\cos (t\log p))/p$.

For an arithmetic function $f$ that is multiplicative and satisfies $|f(n)|\le 1$, defineOne form of Halász theorem gives, for $x,T\ge 2$,Thus a large mean requires small pretentious distance from some Archimedean character. The parameter range matters: $M_f$ minimizes over a bounded interval of real numbers $t$, and the final term accounts for that cutoff.^[ref]

An Archimedean character on the positive integers is the completely multiplicative functionIts modulus is one. The identity $\log (mn)=\log m+\log n$ proves complete multiplicativity, while $t=0$ gives the constant-one arithmetic function. These functions form the comparison family in Halász theorem: a large mean of a bounded multiplicative function can arise when its prime values resemble those of one such character.

For multiplicative functions $f_1,f_2,g_1,g_2$ bounded in modulus by one, the pretentious distance satisfiesHere products mean pointwise products of arithmetic functions. For complex numbers $z,w$ of modulus at most one, write $a=\Re z$ and $b=\Re w$. The bounds $|\Im z|\le \sqrt{2(1-a)}$ and $|\Im w|\le \sqrt{2(1-b)}$ giveApply this with $z=f_1(p)\overline{g_1(p)}$ and $w=f_2(p)\overline{g_2(p)}$ at each prime $p\le x$, divide by $p$, and use the triangle inequality for the resulting finite Euclidean norm. This proves the displayed product inequality.^[ref]

For $a=a_0(1+ϵ)$, equilibrium outer radius $b_0$, and a bubble gas obeying $p_0\pi a^2=\text{constant}$, the linearization isThe bubble therefore undergoes simple harmonic motion with