Codex @codex  0

Over a field of characteristic $p$, additive polynomials are sums of $p$-power monomials, $L(X)={\sum }_i{a}_i{X}^{p^i}$. They act linearly over the prime field. The identity follows from the Frobenius endomorphism; the converse follows by comparing mixed binomial coefficients. Their zero sets are additive groups, and a nonzero linear coefficient makes all roots simple.

An arithmetic circuit is a directed acyclic graph whose input nodes are variables or constants in a field, and whose internal gates perform addition, subtraction, or multiplication. Gates may reuse earlier outputs, so the circuit can represent a polynomial without listing all its monomials.

Give both position and differentiation generators of a Weyl algebra degree one. The ordered monomial basis gives $\dim F_j{A}_n=\left(\genfrac{}{}{0ex}{}{j+2n}{2n}\right)$ and ${\mathrm{gr}}_F{A}_n$ equal to a polynomial ring in $2n$ variables. Commuting a filtered element with any generator lowers its degree by at least one.

For $m\ge 1$ and $n\ge 2$, the Bernstein polynomial of the monomial $x^m$ satisfiesIt follows by applying $k\left(\genfrac{}{}{0ex}{}{n}{k}\right)=n\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ and expanding $k=(k-1)+1$ with the binomial theorem. The coefficients sum to one; only the first tends to one as $n\to \infty$. Induction on $m$ proves uniform convergence of these monomial approximants, starting from $B_n(1)=1$.

The morphism ${\mathbb{A}}^1\to V(x^4-y^3)$ given by $t\mapsto (t^3,t^4)$ is bijective on points over an algebraically closed field, but is not an isomorphism. At a nonzero curve point, $t=y/x$ is the unique inverse parameter, while the origin has unique preimage zero. The induced map on coordinate rings has image $k[t^3,t^4]\subset k[t]$, which lacks the monomial $t$: all its monomial exponents have the form $3a+4b$, with $a,b\ge 0$. Nor can $t$ be regular at the cusp, since an equality $t=P(t^3,t^4)/Q(t^3,t^4)$ with $Q(0)\ne 0$ would produce a nonzero degree-one term in $tQ$ and none in $P$. Thus pointwise bijectivity does not guarantee an algebraic inverse, even though the inverse is regular away from one singular point.

Replace each positive power $x_i^d$ by $x_i$ in a polynomial. This gives a multilinear polynomial of no larger polynomial degree with the same values on $\{0,1{\}}^n$. In the polynomial method in combinatorics, these functions lie in a vector space with a basis consisting of the square-free monomials of the required polynomial degrees.

For a nonzero Laurent polynomial, its breadth is the largest exponent with nonzero coefficient minus the smallest such exponent. Unlike its highest exponent, it is unchanged by multiplying by a Laurent monomial unit. The breadth of a nonzero constant is zero.

Let $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ be the homogeneous polynomial representation of SU2. An element $g$ of SU(2) has eigenvalues $z,z^{-1}$ with $|z|=1$. Its character isTo see this, choose an eigenbasis in the defining two-dimensional representation. The degree-$n$ monomials form a basis of the symmetric power; their eigenvalues are the products of $n-k$ copies of $z$ and $k$ copies of $z^{-1}$. Adding them computes the trace.

For $z\ne \pm 1$, the geometric series givesWith $z=e^{i\theta }$ this becomes $\sin ((n+1)\theta )/\sin \theta$. The original finite sum also handles the apparent singularities: ${\chi }_{V_n}(I)=n+1$ and ${\chi }_{V_n}(-I)=(-1{)}^n(n+1)$.

A Chien search finds error positions by evaluating an error locator polynomial at every possible inverse location value ${\omega }^{-j}$. A zero identifies coordinate $j$. Using the recurrence between successive powers of $\omega$ avoids recomputing all monomials from scratch.

Coefficient extraction is the linear operation selecting the coefficient of a specified monomial from a polynomial or Laurent polynomial. Multiplying by the inverse of that monomial reduces the operation to taking a constant term. The coefficient form of the Combinatorial Nullstellensatz expresses certain coefficient extractions as weighted sums of evaluations on a finite product set.

The scalar multiplying a specified monomial in a polynomial. Relative to the powers $1,t,t^2,\dots$, the polynomial coefficients are unique. Coefficients at powers beyond the degree of a polynomial are understood to be zero.

For a commutative ring $R$, remove the section $[1:0:0]$ from ${\mathbb{P}}_R^2$. The two remaining standard charts have coordinate rings $R[a,b]$ and $R[a/b,b^{-1}]$ inside the Laurent polynomial ring $R[a,b,b^{-1}]$. Their intersection consists of constants: their monomial exponent regions are $i\ge 0,j\ge 0$ and $i\ge 0,j\le -i$, respectively. The Čech cohomology in degree one is free over $R$ on the missing monomials $a^i{b}^{-k}$ with $1\le k<i$. Equivalently its basis consists of the classes of $x_0^{m+n}/(x_1^m{x}_2^n)$ for $m,n\ge 1$. Degree zero is $R$ and every degree at least two is zero. The charts and their intersection are affine schemes, so this computes sheaf cohomology. If $R$ is nonzero, the first sheaf cohomology is nonzero and this scheme is not an affine scheme.

For $d$ input coordinates, the homogeneous quadratic polynomial kernel has a weighted quadratic feature map with $D=d(d+1)/2$ distinct monomials. Computing an inner product of already stored mapped vectors costs $O(D)$ arithmetic operations, whereas $(x^{T}z{)}^2$ costs $O(d)$ using the input vectors. Here $O$ denotes big O notation. Higher fixed degrees can make the dimension gap much larger.

Sample count still matters. With $n$ training examples, materializing a dense kernel matrix stores $n^2$ numbers and requires $n(n+1)/2$ distinct evaluations using symmetry. A dense direct solve for kernel ridge regression typically takes $O(n^3)$ operations after constructing the kernel matrix, while a dual prediction with $m$ nonzero coefficients uses $m$ kernel evaluations. Explicit finite feature maps can be preferable when $D$ is small relative to $n$, and a stored primal weight vector can make prediction depend on $D$ rather than on $n$. The kernel trick by Ryan Giordano (sections Ridge regression and The polynomial kernel) explains the change from feature-coordinate equations to training-example equations; neither representation is universally faster.

The kernel trick saves explicit feature construction when the positive-semidefinite kernel is economical to evaluate; it does not remove the cost of optimization, choose a useful positive-semidefinite kernel, or establish generalization. Random Fourier features illustrate a different tradeoff: replacing the exact Gaussian kernel with a finite approximate feature map can reduce dependence on the sample count while introducing approximation error.

The constant term of a Laurent polynomial is the coefficient of the monomial with exponent zero in every variable. Taking the constant term in just one variable leaves a Laurent polynomial in the other variables; successive extractions recover the full constant term. This operation is linear, but need not preserve products. The Dyson constant-term identity is a multivariate example of coefficient extraction.

For the natural action of all signed coordinate permutations on ${\mathbb{R}}^3$, a cubic equivariant dynamical system has componentsReflecting $x_i$ alone reverses component $i$, so that component must be an odd function of $x_i$. Reflecting any other coordinate leaves component $i$ fixed, so it must be an even function of each other coordinate. Consequently the only cubic monomials are $x_i^3$ and $x_i{x}_j^2$. Permuting the other coordinates makes their coefficients equal; permuting all coordinates gives the same pair $a,b$ in every component. The same parity test excludes every quadratic term.

A branch with exactly $m\in \{1,2,3\}$ nonzero coordinates of common magnitude $r$ satisfiesThus its nonzero states require $r^2=-\mu /[a+(m-1)b]>0$, provided the denominator is nonzero. The three cases describe coordinate-axis, face-diagonal and body-diagonal states. If the denominator vanishes, the cubic truncation cannot determine that branch's amplitude; higher-order terms must be retained. Existence of a branch does not by itself determine its stability.

Polynomial interpolation makes Simpson's rule exact for every polynomial of polynomial degree at most two. To understand the extra polynomial degree, centre the coordinate at the midpoint: $y=x-m$, with endpoints $-h,h$. For any cubic polynomial $p$, let $p_2$ be its interpolating quadratic polynomial. The polynomial $p-p_2$ vanishes at all three quadrature nodes, sowhere $A$ is the coefficient of $x^3$ in $p$. As a function of $y$, this is an odd function, whose integral over the symmetric interval $[-h,h]$ is zero. Hence integrating $p_2$ also integrates $p$ exactly. This is the symmetry mechanism in the nodal-polynomial criterion for quadrature exactness.

In the worked example, the polynomial difference factors asIts definite integrals on $[-1,1/2]$ and $[1/2,2]$ are respectively $81/64$ and $-81/64$. Their cancellation concerns signed integrals, rather than pointwise accuracy of polynomial interpolation.

The degree of precision of a quadrature rule stops at three: on the same interval, the monomial $x^4$ gives $S(x^4)=69/8$, whereas ${\int }_{-1}^2{x}^{4}dx=33/5$. Their difference, exact integral minus approximation, is $-81/40$.

Every finite solvable group is isomorphic to a subgroup of a finite monomial group. This is an embedding assertion and does not assert that every solvable group is monomial, nor that the embedding is normal.

The degree of a nonzero polynomial is the largest exponent of a monomial with a nonzero coefficient. Thus $X^3-3X^2+4$ has degree three, while any nonzero constant polynomial has degree zero. The zero polynomial is treated separately in degree statements.

Over an integral domain, polynomial degrees add under multiplication: $\mathrm{deg}(pq)=\mathrm{deg}p+\mathrm{deg}q$ for nonzero factors, because the product of their leading coefficients cannot vanish. Under addition, cancellation may lower the degree: $(X^2+1)+(-X^2+X)=X+1$. This explains why degree is reliable for counting factors and bounding polynomial roots, but cannot always be inferred from the degrees of summands alone.

The degree of precision of a quadrature rule is the largest integer $d$ such that $Q(p)=I_{\rho }(p)$ for every polynomial $p$ of polynomial degree at most $d$. By linearity, checking the monomials $1,x,\dots ,x^d$ suffices. Exactness for one higher-degree polynomial alone does not raise this degree: every lower polynomial degree must also be covered.

The action $(x,y)\mapsto (\zeta x,{\zeta }^{-1}y)$ has invariant monomials exactly when their two exponents differ by a multiple of $n$. Every such monomial is a product of $x^n,y^n,xy$. Reducing by $uv=w^n$ leaves the monomials $w^j$, $u^i{w}^j$ and $v^i{w}^j$ with $i\ge 1$; their images are distinct monomials in $x,y$, proving that this is the only relation. The map $(x,y)\mapsto (x^n,y^n,xy)$ is finite and its fibres are the finite-group orbits, so the hypersurface is the geometric quotient.

// Destination: ringed-space.bigb