An infinite descent rules out a nonempty set of positive integer solutions by constructing from any solution another with a strictly smaller positive integer measure. Choosing a solution of minimal measure gives a contradiction. Common measures are a hypotenuse, a denominator, or the size of a primitive solution.

For a number field $K$ and fixed nonzero coefficients $\alpha ,\beta \in K$, the equationhas finitely many solutions in the unit group of its ring of integers. The coefficients remain fixed; varying them gives a different finiteness problem.

Here is the logarithmic-form mechanism. The Dirichlet unit theorem writes each unit as a root of unity times a product of fixed fundamental units with integer exponents. If the unit rank is zero there are already only finitely many possibilities. Otherwise set $B$ to the maximum of $3$ and the absolute values of their exponents. The logarithmic embedding and the product formula imply that a unit with a large exponent vector is exponentially small at some archimedean place, after possibly interchanging $u$ and $v$.

At that place, $\alpha u+\beta v=1$ makes the other term exponentially close to one. A nonzero logarithm of that term is a linear form in logarithms of algebraic numbers, including a suitable multiple of $2\pi i$ when complex logarithm branches are adjusted. Its integer coefficients are $O(B)$. The Baker lower bound for a homogeneous linear form in logarithms bounds its nonzero size below by $B^{-C}$ for fixed data, contradicting an exponentially small upper bound when $B$ grows. The logarithm cannot vanish, since then the other term would be exactly one and the remaining nonzero term zero. Thus the exponent vectors are bounded, leaving finitely many solutions. A finite torsion subgroup and finitely many archimedean places allow uniform constants.

An irreducible integral binary form of degree at least three has only finitely many integral solutions at a fixed nonzero value. Factor the form in a splitting field. Each linear factor generates an ideal dividing a fixed ideal, so it is one of finitely many chosen factors times a unit. The linear relation among any three distinct factors gives a unit equation over a number field, hence finitely many ratios and finitely many rational directions. Homogeneity and the fixed nonzero value then bound the scale on each direction. The same argument works for a binary form with at least three distinct projective roots, even without irreducibility.

For every $ϵ>0$, the abc conjecture asserts that there is a constant $K_{ϵ}>0$ such that all coprime positive integers $A+B=C$ satisfyThe radical of an integer is the product of its distinct prime factors. The condition $\gcd (A,B)=1$ makes the three entries pairwise coprime. Repeated prime powers enlarge a number without enlarging its radical in the same way, so the conjecture constrains how often addition can relate unusually concentrated prime factorizations.

The example $(A,B,C)=(1,8,9)$ has $\mathrm{rad}(ABC)=6<9$ and is an abc-hit. Such examples are compatible with the conjecture: its bound includes both the constant $K_{ϵ}$ and the exponent $1+ϵ$. Conditional applications include the primitive Fermat bound from abc and the Catalan power bound from abc.

More generally, fix positive integers $a,b$, a nonzero integer $c$, and positive integers $x,y,n$ satisfying $a{x}^n-b{y}^n=c$. Put $g=\gcd (a{x}^n,b{y}^n)$, which divides $|c|$. Applying the conjecture to the normalized positive triple formed from $a{x}^n$, $b{y}^n$ and $|c|$ gives, for $M=\max (x,y)$,Indeed, the largest normalized entry is at least a fixed multiple of $M^n$, while the radical is at most $\mathrm{rad}(ab|c|)xy\le \mathrm{rad}(ab|c|)M^2$. Taking $ϵ=1/4$ makes the exponent at least $1/2$ for $n\ge 3$, bounding $M$ uniformly; if $M\ge 2$, it also bounds $n$. The separate case $x=y=1$ can satisfy the equation for every $n$ when $a-b=c$.

There are no positive integer solutions to the displayed equation when the exponent is an integer greater than two. Homogeneity means that any hypothetical solution would give infinitely many common multiples; statements bounding solutions by arithmetic height must therefore first distinguish primitive solutions from their multiples.

A Diophantine equation is soluble at a place of a number field if it has a solution over the corresponding completion of a valued field. It is locally soluble everywhere if this holds at all finite and infinite places. Over the rational numbers, these are the real field and all p-adic fields. Local solubility is necessary for a rational point, but the local-to-global gap in isogeny descent shows it need not suffice.

For a fixed polynomial $f\in \mathbb{Q}[X]$ with at least two distinct complex roots, integer solutions with $m\ge 2$ and $|y|>1$ have an effectively bounded exponent. Factorization over a splitting field and the fixed differences between distinct roots control common ideal divisors. Outside a fixed set of primes, each factor's valuation is divisible by $m/\gcd (m,e_i)$, where $e_i$ is its root multiplicity. The generalized power-equation estimates from linear forms in logarithms of algebraic numbers, including unit and exceptional-prime factors, give the exponent bound. A single-root polynomial and the values $y=0,\pm 1$ cannot be included: $4^m=(2^m{)}^2$, and $1^m=f(0)$ for $f(X)=X(X-1)+1$, give infinite exponent families.

Let $K$ be a number field of degree $D$, and let $L_1,\dots ,L_M$ be linear forms in $N>DM$ variables with coefficients in $K$ and projective height at most $\mathcal{H}\ge 1$. There is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ satisfying every $L_j(\mathbf{x})=0$ andFor $K=\mathbb{Q}$, this follows by applying the pigeonhole principle to the images of the integer box $\{0,\dots ,Y{\}}^N$ under $(L_1,\dots ,L_M)$. Expanding coefficients in a rational basis of $K$ gives the factor $D$ in the general count.

For every irrational number $\alpha$ and positive integer $N$, there are integers $p,q$ with $1\le q\le N$ such thatThe proof applies the pigeonhole principle to the $N+1$ fractional parts of $0,\alpha ,\dots ,N\alpha$.

Transcendence theory proves that particular numbers are not algebraic numbers, or that several numbers satisfy no nonzero polynomial relation with algebraic coefficients. A recurring method constructs an auxiliary expression which is simultaneously too small analytically and too large arithmetically if the proposed values were algebraic.

The zero-mode dimensional reduction on a flat $(D-4)$-torus splits a massless p-form gauge field by the number $r$ of spacetime indices. Each sector has $\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)$ copies and $\left(\genfrac{}{}{0ex}{}{2}{r}\right)$ local polarizations per copy. Comparing coefficients in $(1+t{)}^{D-4}(1+t{)}^2$ gives ${\sum }_r\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)\left(\genfrac{}{}{0ex}{}{2}{r}\right)=\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$, proving polarization matching. Four-dimensional three- and four-form potentials contribute no local propagating states.

// Target: supersymmetry.bigb

On a star-shaped neighbourhood of $a$, a smooth function satisfiesThe fundamental theorem of calculus along the segment proves the identity, and differentiation under the integral sign proves smoothness of the coefficients.

A formal pseudodifferential operator is a series in integer powers of $\partial =d/dx$, bounded above but possibly infinite towards negative powers. Its coefficients are smooth functions. Multiplication uses the formal pseudodifferential composition rule. Negative powers are formal inverses, not globally defined integration operators on all smooth functions. Every coefficient in a product is a finite sum, so multiplication is defined without analytic convergence.

For a commutative $k$-algebra $R$, a relative differential operator is a $k$-linear endomorphism. Set $D_k^{-1}(R)=0$, and define order at most $m$ by $[P,m_a]\in D_k^{m-1}(R)$ for every $a\in R$, where $m_a$ means multiplication by $a$. Order zero is multiplication by an element of $R$. The finite-order operators form the ring $D_k(R)$ under composition. Specifying the base is essential: absolute operators on a complex polynomial ring include derivations of the coefficient field and need not be complex-linear.

Suitable curvature functions determine a regular Frenet curve up to a Euclidean rigid motion. In the plane one signed curvature suffices; in three dimensions positive curvature and torsion determine a unit-speed curve, as in the fundamental theorem of regular space curves. The result is local on the parameter interval and assumes the regularity needed by the frame equations.

The Dirichlet energy of a differentiable scalar function is one half the integral of its squared gradient. On a Euclidean domain it is $E(u)=\frac{1}{2}\int |\nabla u{|}^{2}dx$; the Dirichlet energy on a Riemannian manifold uses its metric and volume measure. Its stationary functions satisfy the Laplace equation with fixed boundary values.

Hodge theory connects de Rham cohomology with harmonic differential forms through the Hodge Laplacian on a compact Riemannian manifold. The Hodge decomposition theorem supplies a unique harmonic representative of each cohomology class. On a compact Kähler manifold, the decomposition additionally respects the complex bidegrees of differential forms.

On a graded nilpotent Lie group whose first layer generates its Lie algebra, a horizontal curve obeys ${\dot{\gamma }}_t=(L_{{\gamma }_t}{)}_{\ast }{\dot{x}}_t$ for a control in the first layer. The displayed infimum is over horizontal curves with prescribed endpoints. The distance is left invariant and homogeneous under the graded dilations. On a free nilpotent Lie group it measures the least Euclidean length of a path realizing a prescribed truncated path signature.

Calibrated geometry constructs volume-minimizing submanifolds through closed differential forms with a tangent-plane bound. It gives an algebraic sufficient condition for solving the nonlinear minimal surface problem.

Imagine folding a sheet along its horizontal axis by the smooth map $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$, $F(x,y)=(x,y^2)$. The inputs $(a,r)$ and $(a,-r)$ have the same output. Along $y=0$, horizontal motion still changes the output, but vertical motion produces no first-order change: $F(a,\epsilon )-F(a,0)=(0,{\epsilon }^2)$. These inputs are critical points of a smooth map even though their differential is not the zero linear map. The missing direction in the target tangent space is what matters.

Precisely, let $f:X^m\to Y^n$ be a smooth map between smooth manifolds without boundary. An input $p\in X$ is a critical point of a smooth map when its differentialis not surjective, equivalently when its rank of a linear map is less than $n$. An input where it is surjective is a regular point; a map with no critical points of a smooth map is a submersion. The threshold is the target dimension, not $\min (m,n)$. In manifold charts, compute the matrix rank of the $n\times m$ Jacobian matrix; changes of charts preserve that matrix rank. ^[ref]

For the fold,Thus its set of critical points of a smooth map is the entire horizontal axis. Away from that axis, its differential can produce every infinitesimal target displacement. At the axis, it can produce only horizontal ones. This diagnoses where the two inverse branches in the picture merge.

For a scalar smooth function $f:X\to \mathbb{R}$, the target dimension is one. A linear map to $\mathbb{R}$ fails to be surjective exactly when it is zero, so here a critical point of a smooth map means $d{f}_p=0$. In Euclidean space this is precisely the familiar critical point condition $\nabla f(p)=0$: every first partial derivative vanishes. On a Riemannian manifold, it is equivalently ${\nabla }_{g}f(p)=0$ for the chosen Riemannian metric; the Riemannian gradient depends on that metric, but whether it vanishes does not. See Loring W. Tu, An Introduction to Manifolds, Proposition 8.23 for the scalar coordinate criterion. The scalar smooth function $f(t)=t^3$ has a critical point at zero without a local maximum or local minimum.

A source of lower dimension gives a different warning. For the smooth map $G:\mathbb{R}\to {\mathbb{R}}^2$, $G(t)=(t,t^2)$, the Jacobian matrix is the column $(1,2t{)}^T$, of matrix rank one at every input. Every input is a critical point of a smooth map, since the target dimension is two, even though this parametric curve has nonzero velocity everywhere and is an immersion. More generally, if $m<n$, no differential $T_{p}X\to T_{f(p)}Y$ can be surjective, so every input is critical. The terminology measures the ability to fill target directions, rather than whether the parametrized curve looks smooth.