Dependencies

Alias of MonomialOrder.isGroebnerBasis_iff_isRemainder_sPolynomial_zero.

A basis \(G = \{ g_1, \ldots , g_t \} \) for an ideal \(I\) is a Gröbner basis if and only if \(S(g_i, g_j) \to _G 0\) for all \(i \neq j\).

Alias of MonomialOrder.isGroebnerBasis_iff_isRemainder_sPolynomial_zero.

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 \(G''\).

Let \(G'' \subseteq R[\mathbf{X}]\) be a set of polynomials where every nonzero element has a unit leading coefficient:

Then for any polynomial \(p \in R[\mathbf{X}]\), there exists a remainder \(r\) satisfying:

where \(\operatorname{LC}_m(g)\) denotes the leading coefficient of \(g\) under monomial order \(m\).

Let \(I \subseteq k[x_1, \ldots , x_n]\) be an ideal. Then there exists a finite subset \(G = \{ g_1, \ldots , g_t\} \) of \(I\) such that \(G\) is a Gröbner basis for \(I\).

Remainder of any polynomial on division by Gröbner basis exists and is unique.

Any leading coefficient of polynomial in the Gröbner basis is required to be a unit.

Remainder of any polynomial on division by Gröbner basis exists and is unique.

It is a variant of MonomialOrder.existsUnique_isRemainder_of_isGroebnerBasis, allowing the Gröbner basis to contain also 0, besides polynomials with invertible leading coefficients.

A subset \(s \subseteq R\) has finitely generated span if and only if: \(\exists \) finite \(s' \subseteq s\) such that \(\mathsf{span}(s) = \mathsf{span}(s')\)

Let \(f: \alpha \to \beta \) be a function and \(s \subseteq \alpha \) a subset with finite image \(f(s)\). Then there exists a finite subset \(s' \subseteq _{\text{fin}} s\) such that:

• \(s' \subseteq s\) (subset relation)

• \(f(s') = f(s)\) (image equality)

• \(|s'| = |f(s)|\) (cardinality preservation)

Let \(G = \{ g_1, \ldots , g_t\} \) be a Gröbner basis for an ideal \(I \subseteq k[x_1, \ldots , x_n]\). Then \(G\) is a basis for the vector space \(I\) over \(k\).

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

Using the convention that \(\langle \emptyset \rangle = \{ 0\} \), we define the empty set \(\emptyset \) to be the Gröbner basis of the zero ideal \(\{ 0\} \).

A basis \(G = \{ g_1, \ldots , g_t \} \) for an ideal \(I\) is a Gröbner basis if and only if \(S(g_i, g_j) \to _G 0\) for all \(i \neq j\).

A finite set of polynomials is a Gröbner basis of an ideal if and only if it is a subset of this ideal and 0 is a remainder of each member of this ideal on division by this finite set.

Any leading coefficient of polynomial in the finite set is required to be a unit.

A finite set of polynomials is a Gröbner basis of an ideal if and only if it is a subset of this ideal and 0 is a remainder of each member of this ideal on division by this finite set.

It is a variant of MonomialOrder.isGroebnerBasis_iff_subset_ideal_and_isRemainder_zero, over a field and without hypothesis on leading coefficients in the finite set.

Formally:

Let \(G = \{ g_1, \ldots , g_t\} \) be a finite subset of \(k[x_1, \ldots , x_n]\). Then \(G\) is a Gröbner basis for the ideal \(I = \langle G \rangle \) if and only if for every \(f \in I\), the remainder of \(f\) on division by \(G\) is zero. whose leading coefficients are invertible with respect to a monomial order

A finite set of polynomials is a Gröbner basis of an ideal if and only if it is a subset of this ideal and 0 is a remainder of each member of this ideal on division by this finite set.

It is a variant of MonomialOrder.isGroebnerBasis_iff_subset_ideal_and_isRemainder_zero, allowing the finite set to contain also 0, besides polynomials with invertible leading coefficients.

Given a multivariate polynomial \(f\) 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

1. \(f = \sum _{b\in B} g(b)b + r\)

2. degree of any non-zero \(g(b)b\) where bB is less than or equal to the degree of \(p\),

3. 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.

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 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.

The leading term in a multivariate polynomial is zero if and only if this polynomial is zero.

Let \(R\) be a commutative ring, and let \(G'' \subseteq R[x_1, \dots , x_n]\), \(I \subseteq R[x_1, \dots , x_n]\) be an ideal, and \(p, r \in R[x_1, \dots , x_n]\). Assume that:

• \(G'' \subseteq I\),

• \(r\) is the remainder of \(p\) upon division by \(G''\).

Then,

Let \(G'' \subseteq R[x_1, \dots , x_n]\), let \(I \subseteq R[x_1, \dots , x_n]\) be an ideal, and let \(p, r \in R[x_1, \dots , x_n]\). Suppose that:

• \(G'' \subseteq I\),

• \(r \in I\),

• \(r\) is the remainder of \(p\) upon division by \(G''\).

Then,

Given a remainder r of a polynomial p on division by a Gröbner basis G of an ideal I, the remainder r is 0 if and only if p is in the ideal I.

Any leading coefficient of polynomial in the Gröbner basis G is required to be a unit.

Formally:

Let \(G = \{ g_1, \dots , g_t\} \) be a Gröbner basis for an ideal \(I \subseteq k[x_1, \dots , x_n]\) and let \(f \in k[x_1, \dots , x_n]\). Then \(f \in I\) if and only if the remainder on division of \(f\) by \(G\) is zero.

Given a remainder r of a polynomial p on division by a Gröbner basis G of an ideal I, the remainder r is 0 if and only if p is in the ideal I.

It is variant of MonomialOrder.remainder_eq_zero_iff_mem_ideal_of_isGroebner, over a field and without hypothesis on leading coefficients in the finite set.

Given a remainder r of a polynomial p on division by a Gröbner basis G of an ideal I, the remainder r is 0 if and only if p is in the ideal I.

It is a variant of MonomialOrder.remainder_eq_zero_iff_mem_ideal_of_isGroebner, allowing the finite set to contain also 0, besides polynomials with invertible leading coefficients.

Let \(G'' \subseteq R[x_1, \dots , x_n]\) be a set of polynomials such that

Then,

The \(S\)-polynomial of \(f\) and \(g\) is the combination

the S-polynomial of \(f\) and \(g\) is antisymmetric:

Let \(f, h_1, \dots , h_m \in k[\mathbf{x}] \setminus \{ 0\} \), and suppose

If

then

Furthermore, if \(S(h_i, h_j) \ne 0\), then \(\mathrm{lm}(h_i) {\gt} \mathrm{lm}(S(h_i, h_j))\).

For any polynomial \(g \in \operatorname{MvPolynomial}{\sigma }{R}\) and monomial order \(m\), the S-polynomial with zero as first argument vanishes:

\(h_1, h_2 \in k[\mathbf{x}], lm(h_1) = lm(h_2), S(h_1, h_2) \ne 0\), then \(lm(S(h_1, h_2)) {\lt} lm(h_1)\).

For any polynomial \(g \in \operatorname{MvPolynomial}{\sigma }{R}\) and monomial order \(m\), the S-polynomial with zero as second argument vanishes:

Let \(I \subseteq k[x_i : i \in \sigma ]\) be an ideal, and let \(G \subseteq I\) be a finite subset. Suppose that \(r_1\) and \(r_2\) are generalized remainders of a polynomial \(p\) upon division by \(G\). Then,