Dependencies

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 \(i \neq j\).

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 \(k\) 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 \(I \subseteq k[x_1, \ldots , x_n]\) 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 \(I\).

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 \(G = \{ g_1, \dots , g_t\} \) 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.

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

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 \(\{ 0\} \).

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.

Fix a monomial order \({\gt}\) on \(\mathbb {Z}_{\geq 0}^n\), and let \(F = (f_1, \ldots , f_s)\) be an ordered \(s\)-tuple of polynomials in \(k[x_1, \ldots , x_n]\). Then every \(f \in k[x_1, \ldots , x_n]\) can be written as

where \(a_i, r \in k[x_1, \ldots , x_n]\), and either \(r = 0\) or \(r\) is a linear combination, with coefficients in \(k\), of monomials, none of which is divisible by any of \(\mathrm{LT}(f_1), \ldots , \mathrm{LT}(f_s)\). We will call \(r\) a remainder of \(f\) on division by \(F\).

Let \(p \in R[\mathbf{X}]\), \(G'' \subseteq R[\mathbf{X}]\) be a set of polynomials, and \(r \in R[\mathbf{X}]\). Then \(r\) is a remainder of \(p\) modulo \(G''\) with respect to monomial order \(m\) if and only if there exists a finite linear combination from \(G''\) such that:

1. The support of the combination is contained in \(G''\)

2. \(p\) decomposes as the sum of this combination and \(r\)

3. For each \(g' \in G''\), the degree of \(g' \cdot (coefficient\ of\ g')\) is bounded by \(\deg _m(p)\)

4. No term of \(r\) is divisible by any leading term of non-zero elements in \(G''\)

Let \(p, r \in k[x_i : i \in \sigma ]\), and let \(G' \subseteq k[x_i : i \in \sigma ]\) be a finite set. We say that \(r\) is a generalized remainder of \(p\) upon division by \(G'\) if the following two conditions hold:

1. For every nonzero \(g \in G'\) and every monomial \(x^s \in \operatorname {supp}(r)\), there exists some component \(j \in \sigma \) such that

   \[ \operatorname {multideg}(g)_j {\gt} s_j. \]

2. There exists a function \(q : G' \to k[x_i : i \in \sigma ]\) such that:

- For every $g \in G'$,
      $$
      \operatorname{multideg}''(q(g)g) \leq \operatorname{multideg}''(p);
      $$
- The decomposition holds:
      $$
      p = \sum_{g \in G'} q(g)g + r.
      $$

Let \(p \in R[\mathbf{X}]\) be a polynomial, \(G'' \subseteq R[\mathbf{X}]\) a set of polynomials, and \(r \in R[\mathbf{X}]\) a remainder. Then the remainder property is invariant under inserting the zero polynomial:

Let \(p \in R[\mathbf{X}]\) be a polynomial, \(G'' \subseteq R[\mathbf{X}]\) a set of polynomials, and \(r \in R[\mathbf{X}]\) a remainder. Then the remainder property is invariant under removal of the zero polynomial:

Given a nonzero polynomial \(f \in k[x]\), let

where \(c_i \in k\) and \(c_0 \neq 0\) [thus, \(m = \deg (f)\)]. Then we say that \(c_0 x^m\) is the leading term of \(f\), written

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

Then,

For any set of polynomials \(G'' \subseteq R[\mathbf{X}]\) and monomial order \(m\), the image of leading terms on the nonzero elements of \(G''\) equals the image on all elements minus zero:

Let \(p \in R[\mathbf{X}]\) be a multivariate polynomial. Then the leading term of \(p\) vanishes with respect to monomial order \(m\) if and only if \(p\) is the zero polynomial:

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,

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 \(I \subseteq k[x_i : i \in \sigma ]\) 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,

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 \(k\).

For any subset \(s \subseteq R\) of a ring \(R\), inserting zero does not change the linear span:

For any subset \(s \subseteq R\) of a ring \(R\), removing zero does not change the linear span:

For any ring \(R\), the span of the zero singleton set equals the zero submodule:

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 \(\mathrm{lm}(h_i) {\gt} \mathrm{lm}(S(h_i, h_j))\).

\(h_1, h_2 \in k[\mathbf{x}], lm(h_1) = lm(h_2), S(h_1, h_2) \ne 0\), 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 first argument vanishes:

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

a Partially Ordered Set, a ≥0