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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 $B_n(1)=1$.

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.

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.

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.

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 $-81/40$.

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

The monomial $x^i{y}^j$ is an eigenvector of the differential linear operator $x^2{\partial }_x^2+y^2{\partial }_y^2$, with eigenvalue $i(i-1)+j(j-1)$. Over the real or complex numbers this vanishes exactly for $i,j\in \{0,1\}$. On polynomials of bounded total degree, the kernel is therefore spanned by the permitted members of $1,x,y,xy$, while the image of a linear map is spanned by all remaining monomials. The trace is the sum of those diagonal eigenvalues.

The vector space of formal polynomials over a field $k$ in $n$ variables with total degree of a polynomial at most $d$ has the monomial basis $x_1^{a_1}\cdots x_n^{a_n}$ for nonnegative exponents with ${\sum }_i{a}_i\le d$. Introducing a slack exponent and using stars and bars counts $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ basis elements. These are formal polynomials; over a finite field, distinct formal polynomials need not define distinct polynomial functions unless suitable degree restrictions hold.

For two binary operations, multiplication distributes over addition on the left or right when the respective displayed identity holds. If multiplication is commutative, either law gives the other. Repeated distribution expands expressions in an incline into finite sums of monomials. This property is distinct from associativity of either operation.

The dual of $D_{\lambda }$ has dominant label $-w_0\lambda =(-{\lambda }_m,\dots ,-{\lambda }_1)$. Substitution $x_i\mapsto x_i^{-1}$ in the alternant character formula for the general linear group and reversal of determinant columns gives that character. The numerator and denominator reversal signs and common monomial factors cancel.

There is a constant $C<3$ such that every cap set in ${\mathbb{F}}_3^n$ has cardinality less than $C^n$.

For the polynomial method in combinatorics, useOver ${\mathbb{F}}_3$, this is the indicator function of $x+y+z=0$. On $A^3$ it is therefore a diagonal tensor with $|A|$ nonzero diagonal entries. Every monomial in its expansion has individual exponents at most two and total degree at most $2n$, so one of its three variable blocks has degree at most $2n/3$. Grouping terms according to such a block and using the slice rank of a diagonal tensor giveswhere $m_n$ is the number of $\alpha \in \{0,1,2{\}}^n$ with ${\sum }_i{\alpha }_i\le 2n/3$. If $X_i=1-{\alpha }_i$ are independent random variables uniform on $\{-1,0,1\}$, thenAny exponential upper bound for this tail probability gives $m_n\le (3-ϵ{)}^n$ and hence the result after absorbing the factor three and finitely many small dimensions into $C<3$.