0 is less then any σ →₀ ℕ w.r.t. monomial order.
The leading term in a non-zero multivariate polynomial is the term of the polynomial's degree in the polynomial. The leading term in the zero polynomial is defined as the zero polynomial.
Equations
- m.leadingTerm f = (MvPolynomial.monomial (m.degree f)) (m.leadingCoeff f)
Instances For
Given a multivariate polynomial $p$ and a set $B$ of multivariate polynomials over a commutative semiring $R$, and a monomial order. If there exists a $g : B \to R[X]$ with finite support, and a multivriate polynomial $r$, such that
- $p = \sum_{b\in B} g(b)b + r,$
- degree of any non-zero $g(b)b$ where b\in B is less than or equal to the degree of $p$,
- none of terms of $r$ is divisible by leading monomial of a non-zero elements of $B$,
then $r$ is called a remainder of $p$ on division by "divisors" $B$.
A statement that r is a remainder of p on division by B w.r.t. a monomial order m is
denoted as m.IsRemainder p B r.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A variant of IsRemainder without coercion of a Set (MvPolynomial σ R).
A variant of IsRemainder where g : MvPolynomial σ R →₀ MvPolynomial σ R is replaced with a
function g : MvPolynomial σ R → MvPolynomial σ R without limitation on its support.
Remainders are preserved on insertion of the zero polynomial into the set of divisors.
Remainders are preserved with the zero polynomial removed from the set of divisors.
The leading term in a multivariate polynomial is zero if and only if this polynomial is zero.
Fix a monomial order on the polynomial ring $k[x_1, \ldots, x_n]$.A finite subset $G = \{g_1, \ldots, g_t\}$ of an ideal $I \subseteq k[x_1, \ldots, x_n]$, with $I \ne \{0\}$, is said to be a Gröbner basis (or standard basis) if $$ \langle \operatorname{LT}(g_1), \ldots, \operatorname{LT}(g_t) \rangle = \langle \operatorname{LT}(I) \rangle. $$ Using the convention that $\langle \emptyset \rangle = \{0\}$, we define the empty set $\emptyset$ to be the Gröbner basis of the zero ideal $\{0\}$.
Equations
- m.IsGroebnerBasis G I = (↑G ⊆ ↑I ∧ Ideal.span (m.leadingTerm '' ↑I) = Ideal.span (m.leadingTerm '' ↑G))
Instances For
The $S$-polynomial of $f$ and $g$ is the combination $$ S(f, g) = \frac{x^\gamma}{\mathrm{LT}(f)} \cdot f - \frac{x^\gamma}{\mathrm{LT}(g)} \cdot g. $$
Equations
- m.sPolynomial f g = (MvPolynomial.monomial (m.degree g - m.degree f)) (m.leadingCoeff g) * f - (MvPolynomial.monomial (m.degree f - m.degree g)) (m.leadingCoeff f) * g
Instances For
the S-polynomial of $f$ and $g$ is antisymmetric: $$ \Sph{f}{g} = -\Sph{g}{f} $$
For any polynomial $g \in \MvPolynomial{\sigma}{R}$ and monomial order $m$, the S-polynomial with zero as first argument vanishes: $$ \Sph{0}{g} = 0 $$
For any polynomial $g \in \MvPolynomial{\sigma}{R}$ and monomial order $m$, the S-polynomial with zero as second argument vanishes: $$ \Sph{f}{0} = 0 $$
Let $G'' \subseteq R[\mathbf{X}]$ be a set of polynomials where every nonzero element has a unit leading coefficient: $$ \forall g \in G'',\ \big(\mathrm{IsUnit}(\LC_m(g)) \lor g = 0\big) $$ Then for any polynomial $p \in R[\mathbf{X}]$, there exists a remainder $r$ satisfying: $$ \mathsf{IsRemainder}_m\,p\,G''\,r $$ where $\LC_m(g)$ denotes the leading coefficient of $g$ under monomial order $m$.
$h_1, h_2 \in k[\mathbf{x}], lm(h_1) = lm(h_2), S(h_1, h_2) \ne 0$, then $lm(S(h_1, h_2)) < lm(h_1)$.
Monomial degree of product
Let $k$ be a field, and let $G'' \subseteq k[x_i : i \in \sigma]$ be a set of polynomials. Then for any $p \in k[x_i : i \in \sigma]$, there exists a generalized remainder $r$ of $p$ upon division by $G''$.