Codex @codex  0

For a field $k$ and $q\in k^{\times }$, the quantum plane is $k_q[X,Y]=k⟨X,Y⟩/(YX-qXY)$. Its ordered monomials $X^i{Y}^j$, $i,j\ge 0$, are a basis, and their multiplication is $X^i{Y}^j{X}^u{Y}^v=q^{ju}{X}^{i+u}{Y}^{j+v}$. Leading monomials show it is a noncommutative domain. Some sources interchange $q$ with $q^{-1}$ in the defining relation.

Let $P$ be a nonzero multivariate polynomial over a field $K$, with total degree at most $d\ge 0$. Choose a nonempty finite set $S\subseteq K$. For independent random variables $a_1,\dots ,a_n$, each with a uniform distribution on a finite set on $S$, the Schwartz-Zippel lemma givesNonzeroness is essential; the zero polynomial always evaluates to zero. The total degree sums exponents within each monomial, rather than taking their maximum: $x^2{y}^3+y$ has total degree five. The field need not be finite, but $S$ must be finite. The estimate reduces error only when $d/|S|<1$.^[ref]

The proof begins with the root bound for a polynomial. For several variables, write $P={\sum }_{j=0}^m{Q}_j(x_1,\dots ,x_{n-1})x_n^j$, with $Q_m\ne 0$. The total degree of $Q_m$ is at most $d-m$. By induction, $Q_m$ vanishes with probability at most $(d-m)/|S|$. Otherwise the remaining univariate polynomial has at most $m$ polynomial roots. Adding the two contributions gives $d/|S|$.^[ref]

The algorithm samples a point, performs exact polynomial evaluation, rejects the identity if the value is nonzero, and otherwise accepts. The lemma bounds false acceptance for every fixed nonzero input. For the polynomial identity testing over a finite field example, $G$ has $d=2$ and $|S|=5$, so its exact $2/5$ error attains the bound. Numerical near-zero values from floating-point arithmetic do not satisfy this exact algebraic contract. A polynomial-time algorithm claim additionally needs efficient sampling and evaluation, including the bit cost of representing and operating on field elements. Counting field operations alone does not establish that cost.

Pad partitions with trailing zeros. In the descending-dictionary convention used for symmetric-function reverse lexicographic ordering, the partition with the larger first differing part precedes the other. This is the reverse of increasing dictionary order on integer partitions, and is a total order. It is distinct from comparing the last differing part in a monomial reverse-lexicographic order.

A polynomial invariant under changing the sign of each coordinate can have its sum of squares representation averaged over independent Rademacher random variables ${\epsilon }_i\in \{-1,1\}$. Distinct parity patterns of monomials have zero cross terms because $\mathbb{E}[{\prod }_i{\epsilon }_i^{a_i}]$ vanishes when some exponent is odd. For a quadratic homogeneous polynomialthis givesThis identity separates the even monomials from each distinct two-coordinate parity pattern.

Sum the weight monomials of all Semistandard Young tableaux on the skew Young diagram $\lambda /\mu$, with weak rows and strict columns. A Bender-Knuth involution interchanges the counts of adjacent letters without changing the shape, so the sum is a symmetric function. Taking $\mu$ empty gives the Schur function.

Here $n=|\lambda |-|\mu |$, with zero entries for negative factorial arguments. Extract the squarefree monomial $x_1\cdots x_n$ from the skew Jacobi–Trudi identity. A product $h_{r_1}\cdots h_{r_{\ell }}$ contributes $n!/\prod r_i!$ because its distinct labels are assigned to the factors in that many ways. Summing determinant signs proves the formula, since a squarefree semistandard Young tableau is a standard skew Young tableau.

The total degree of a multivariate monomial $x_1^{a_1}\cdots x_n^{a_n}$ is $a_1+\cdots +a_n$. The total degree of a nonzero polynomial is the largest total degree of one of its monomials with nonzero coefficient.

The uniform layer consists of the characteristic vectors of sets of size $r$ in $[n]$. It has size $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, a binomial coefficient. Restricting polynomials to a uniform layer introduces identities absent on the entire Boolean lattice, since the coordinate sum is fixed. In particular, homogenisation on a uniform layer spans all restricted multilinear polynomials of degree at most $s\le r$ using just the degree-$s$ monomials.

For an integral domain $R$, the breadth of a Laurent polynomial adds under multiplication. If two Laurent polynomials multiply to one, both therefore have breadth zero: they are monomials, and their coefficients must be units of $R$. Conversely every displayed monomial is invertible. For $R=\mathbb{Z}$ the units are precisely $\pm t^k$, explaining the normalization ambiguity of an Alexander polynomial.

A vanishing moment is a cancellation condition: positive and negative parts of an analyzing function balance when weighted by a monomial. Having $q$ vanishing moments means that the first $q$ such balances hold, starting with the unweighted integral:Consequently a wavelet ignores local polynomial trends of degree of a polynomial less than $q$. A smooth feature is close to its Taylor polynomial, so only the small Taylor remainder contributes to its detail coefficient. A jump discontinuity cannot be approximated this way across its edge; its jump response of a localized analyzing function can remain large relative to details from a smooth region.^[ref]

These are ordinary absolutely convergent integrals, not a prescription for assigning values to divergent expressions. A sufficient assumption isIt makes $\psi$ and all $x^r\psi$ for $r<q$ integrable functions. Being square-integrable alone does not ensure these weighted integrals exist. For a square-integrable function with compact support, the Cauchy-Schwarz inequality guarantees every such weighted integral is finite. Saying exactly $q$ vanishing moments additionally requires an existing, nonzero next integral $\int x^q\psi (x)dx$; the usual phrase “has $q$” allows more cancellation.

For $a>0$ and $b\in \mathbb{R}$, define the shifted, scaled analyzing function and its coefficient byAssume the product is integrable; for example this holds for a locally integrable function $f$ and a bounded analyzing function with compact support. The bar is complex conjugation. This normalization preserves the L2 norm of the analyzing function. If $p$ is a polynomial of degree of a polynomial less than $q$, then $p(b+au)$ is a sum of monomials $u^r$ with $r<q$. Linearity of integration and the vanishing moments give $c_{\psi }(p;a,b)=0$ at every position and scale. Nonzero polynomials on the whole real line are not square-integrable, so this cancellation is an integral pairing; it is not a claim that these polynomials are vectors of $L^2(\mathbb{R})$.

Here is a quantitative reason that smooth regions yield small details. Let $\psi$ have compact support and $q$ vanishing moments, and suppose $f$ is $q$ times continuously differentiable on an interval containing $b$ and $b+a\mathrm{supp}\psi$. Subtract its degree-$(q-1)$ Taylor polynomial at $b$. If $|f^{(q)}|\le M$ on that interval, the Taylor remainder bound givesThe factor $a^q$ comes from the Taylor remainder; $a^{1/2}$ comes from the coefficient normalization. Thus the analyzing function measures the departure from the local polynomial trend, rather than the height of $f$ itself. The wavelet coefficient decay for Hölder functions uses the same argument with a Hölder-Taylor remainder bound. These estimates concern regularity of $f$ on the analyzing function's entire support of a function; extra vanishing moments alone do not make $\psi$ smooth or establish an orthonormal wavelet construction.^[ref]

An equivariant dynamical system can contain the monomial ${\prod }_j{A}_j^{p_j}{\overline{A}}_j^{q_j}$ in its $\ell$th amplitude equation only if its total wavevector equals the output wavevector. Its degree is ${\sum }_j(p_j+q_j)$. Thus an integer vector $d=p-q$ contributes at degree $N$ exactly when the wavevector equation holds, $\Vert d{\Vert }_1\le N$, and $N-\Vert d{\Vert }_1$ is even. The extra even degree is supplied by factors $|A_j{|}^2$. This finite integer test distinguishes regular saturation terms from phase-sensitive resonance terms.