Gröbner Bases and Elimination Theory

Introduction

The polynomial ring $K[x_1,\ldots,x_n]$ over a field is Noetherian by Noetherian and Artinian Rings, so every ideal in it is finitely generated; but the two questions one wants to answer about a finitely generated ideal — whether a given polynomial lies in it, and what its common zeros are — have no direct answer from the generators alone, because division of a polynomial by several polynomials leaves a remainder which depends on the order in which the divisions are performed. Gröbner bases remove the ambiguity. Once a monomial order is fixed, a finite set $G$ of generators of an ideal $I$ can be chosen with the property that the remainder on division by $G$ is unique, and then membership in $I$ is decided by a division, the elimination ideals $I \cap K[x_{k+1},\ldots,x_n]$ are read off from $G$, and a system of polynomial equations can be solved by a triangularisation.

The theory is entirely algebraic — monomial orders are well-orderings of a free commutative monoid, division is the Euclidean division of multivariate polynomials, and the algorithm of Buchberger is a finite computation — and it belongs to this Part because it is the computational form of the ideal theory of the commutative ring. It is the effective counterpart of the theorems recorded in Noetherian and Artinian Rings, Integral Extensions and Krull Dimension and Primary Decomposition, and the tool that makes concrete the intersection theory of Algebraic Curves and the elimination performed, both. The Hilbert function and the syzygies of a module over a polynomial ring, which refine the theory, and the geometry of the varieties computed here, belong to the Part II agent who owns.

Throughout, $K$ is a field — the algorithms are most often run over $\mathbb{Q}$ or $\mathbb{F}_p$, and the results are stated for a general field with the characteristic named where it matters — $R = K[x_1,\ldots,x_n]$ is the polynomial ring, $I$ and $J$ are ideals of $R$, and monomial means an element of $R$ of the form $x^\alpha = x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ with $\alpha \in \mathbb{N}^n$. For a nonzero polynomial $f$, $\operatorname{LT}(f)$, $\operatorname{LM}(f)$ and $\operatorname{LC}(f)$ denote its leading term, leading monomial and leading coefficient with respect to a fixed monomial order; for an ideal $I$, $\operatorname{LT}(I)$ is the ideal generated by the leading terms of its nonzero elements.

Monomial Orders and Division

Monomial Orders

Definition. A monomial order on $R$ is a total order $\succ$ on the set of monomials such that

(a) $1 \prec x^\alpha$ for every $\alpha \neq 0$, and

(b) $x^\alpha \succ x^\beta$ implies $x^{\alpha+\gamma} \succ x^{\beta+\gamma}$ for all $\alpha,\beta,\gamma$.

Equivalently, $\succ$ is a well-ordering of the monomials compatible with multiplication; condition (a) and (b) together are equivalent to $\succ$ being a well-ordering, by Dickson's lemma below.

Definition. The three standard orders, for exponent vectors $\alpha, \beta \in \mathbb{N}^n$ with $\lvert\alpha\rvert = \sum\alpha_i$:

lexicographic, $x^\alpha \succ_{\mathrm{lex}} x^\beta$ if the leftmost nonzero entry of $\alpha - \beta$ is positive;
graded lexicographic, $x^\alpha \succ_{\mathrm{grlex}} x^\beta$ if $\lvert\alpha\rvert > \lvert\beta\rvert$, or $\lvert\alpha\rvert = \lvert\beta\rvert$ and $x^\alpha \succ_{\mathrm{lex}} x^\beta$;
graded reverse lexicographic, $x^\alpha \succ_{\mathrm{grevlex}} x^\beta$ if $\lvert\alpha\rvert > \lvert\beta\rvert$, or $\lvert\alpha\rvert = \lvert\beta\rvert$ and the rightmost nonzero entry of $\alpha - \beta$ is negative.

Each is a monomial order. For $n = 2$ and $x \succ y$: $\mathrm{lex}$ orders monomials by degree in $x$ first, so $x \succ y^5$; grevlex orders first by total degree, so $y^5 \succ x$.

Lemma (Dickson). Every set $S$ of monomials in $R$ has a finite subset $S_0$ such that every element of $S$ is divisible by some element of $S_0$; equivalently, the monomial ideals of $R$ are finitely generated.

Proof. For $n = 1$ the assertion is that a nonempty set of nonnegative integers has a least element. For $n > 1$ write the monomials as $x_n^{a}\cdot(\text{monomial in }n-1\text{ variables})$ and induct on $n$: the set of exponents $a$ occurring is a set of nonnegative integers with a least element $a_0$, the monomials of $S$ with $x_n$-exponent $a_0$ are handled by the induction hypothesis in $n-1$ variables, and each of the finitely many integers $a < a_0$ contributes finitely many generators. Alternatively, and this is the standard form for ideal theory, the argument shows that $R$ is Noetherian by a direct proof on the monomial ideals, which one then lifts to all ideals by the same division argument as below.

Division with Remainder

Theorem (division algorithm). Fix a monomial order on $R$ and let $F = (f_1,\ldots,f_s)$ be an ordered $s$-tuple of nonzero polynomials. Then every $f \in R$ can be written

$$ f = q_1f_1 + \cdots + q_sf_s + r , $$

with $q_i, r \in R$ and with $r$ such that no monomial of $r$ is divisible by any of $\operatorname{LM}(f_1),\ldots,\operatorname{LM}(f_s)$. The remainder $r$ is produced by the following finite procedure: while $f \neq 0$, if $\operatorname{LM}(f)$ is divisible by some $\operatorname{LM}(f_i)$ replace $f$ by $f - \frac{\operatorname{LT}(f)}{\operatorname{LT}(f_i)}f_i$, otherwise move $\operatorname{LT}(f)$ into the remainder and delete it from $f$.

Example. In $K[x,y]$ with lex $x \succ y$, divide $f = x^2y + xy^2 + y^3$ by $F = (f_1, f_2)$ with $f_1 = xy - 1$ and $f_2 = y^2 - x$. The leading monomials are $xy$ and $y^2$; the leading term of $f$ is $x^2y$, divisible by $xy$, so subtract $x\cdot f_1 = x^2y - x$ and obtain $xy^2 + y^3 + x$; its leading term $xy^2$ is divisible by $xy$, subtract $y\cdot f_1 = xy^2 - y$, and obtain $y^3 + x + y$; the leading term is now $x$ — under lex a monomial involving $x$ exceeds every monomial in $y$ alone — which no leading monomial divides, so $x$ passes to the remainder and the leading term becomes $y^3$, divisible by $y^2$: subtract $y\cdot f_2 = y^3 - xy$, obtaining $xy + y$; the leading term $xy$ is divisible by $xy$, subtract $f_1 = xy - 1$, obtaining $y + 1$, whose leading term $y$ passes to the remainder, as does the final constant; so the remainder is $r = x+y+1$. Division of the same $f$ by the permuted tuple $(f_2,f_1)$ gives a different remainder, which is what Gröbner bases are designed to prevent.

Gröbner Bases

Definition and Uniqueness

Definition. Fix a monomial order on $R$ and let $I \subseteq R$ be a nonzero ideal. A finite set $G = \{g_1,\ldots,g_t\} \subseteq I$ is a Gröbner basis of $I$ if

$$ \langle \operatorname{LT}(g_1), \ldots, \operatorname{LT}(g_t) \rangle = \operatorname{LT}(I) , $$

equivalently if $\{g_1,\ldots,g_t\}$ generates $I$ and for every nonzero $f \in I$ some $\operatorname{LM}(g_i)$ divides $\operatorname{LM}(f)$; equivalently if the remainder of any polynomial on division by $G$ does not depend on the order of $G$ and vanishes exactly on $I$. The basis is reduced if $\operatorname{LC}(g_i) = 1$ for all $i$ and no monomial of $g_i$ lies in $\langle \operatorname{LM}(g_j) : j \neq i\rangle$.

Theorem (Hilbert basis theorem, sharpened). Let $I$ be a nonzero ideal of $R = K[x_1,\ldots,x_n]$. Then $I$ has a Gröbner basis with respect to every monomial order, $I$ is finitely generated, and the reduced Gröbner basis of $I$ with respect to a fixed monomial order is unique.

Proof. The monomial ideal $\operatorname{LT}(I)$ is generated by the monomials $\operatorname{LM}(f)$ for $f \in I$; by Dickson's lemma it is generated by finitely many of them, say $\operatorname{LM}(g_1),\ldots,\operatorname{LM}(g_t)$ with $g_i \in I$. Then $\langle \operatorname{LT}(g_1),\ldots,\operatorname{LT}(g_t)\rangle = \operatorname{LT}(I)$ by construction, and $G = \{g_1,\ldots,g_t\}$ is a Gröbner basis; dividing any $g \in I$ by $G$ gives a remainder in $I$ whose monomials are divisible by no $\operatorname{LM}(g_i)$, hence a remainder $0$, so $G$ generates $I$. Uniqueness of the reduced basis: if $G$ and $G'$ are reduced Gröbner bases of the same ideal, every element of $G'$ has a leading monomial in the monomial ideal $\operatorname{LT}(I)$, hence divisible by some $\operatorname{LM}(g)$ with $g \in G$, and symmetrically; reducing $g' \in G'$ modulo $G$ and using the two reducedness conditions forces $g' = g$.

Theorem (ideal membership). Fix a monomial order, let $G$ be a Gröbner basis of $I$, and let $f \in R$. Then $f \in I$ if and only if the remainder of $f$ on division by $G$ is zero. Consequently $G$ decides membership in $I$, the equality of two ideals presented by generators (compute reduced Gröbner bases and compare), and the generation of $I$ by a given set.

Buchberger's Algorithm

Definition. For nonzero $f, g \in R$ let $x^\gamma = \operatorname{lcm}(\operatorname{LM}(f),\operatorname{LM}(g))$. The $S$-polynomial is

$$ S(f,g) = \frac{x^\gamma}{\operatorname{LT}(f)}f - \frac{x^\gamma}{\operatorname{LT}(g)}g , $$

a combination of $f$ and $g$ whose leading terms cancel.

Theorem (Buchberger's criterion). A finite set $G$ generating an ideal $I$ is a Gröbner basis of $I$ if and only if for all pairs $g_i, g_j \in G$ the remainder of $S(g_i,g_j)$ on division by $G$ is zero.

Theorem (Buchberger's algorithm). Given a finite set $F$ generating $I$, the following procedure terminates and produces a Gröbner basis of $I$: start with $G = F$; while some pair $(g_i,g_j)$ has nonzero remainder $r$ of $S(g_i,g_j)$ on division by $G$, adjoin $r$ to $G$; then reduce, that is, discard any element whose leading monomial is divisible by the leading monomial of another and replace each remaining element by its monic normal form, obtaining the reduced Gröbner basis.

Proof sketch. Termination follows from Dickson's lemma: the ideal of leading terms of the successive $G$'s is contained in $\operatorname{LT}(I)$ and strictly increases with each genuine enlargement, while monomial ideals are finitely generated and have no infinite strictly increasing chain. Correctness follows from the criterion: when the algorithm stops, every remainder is zero, hence $G$ is a Gröbner basis of the ideal it generates, which is $I$ since $G \supseteq F$ and every adjoined element is a combination of the previous ones.

Example. Let $I = \langle xy-1,\ x^2-y\rangle$ in $K[x,y]$ with lex $x \succ y$. The leading monomials are $xy$ and $x^2$, and

$$ S(xy-1, x^2-y) = x(xy-1) - y(x^2-y) = y^2 - x , $$

whose leading monomial $x$ is divisible by neither $xy$ nor $x^2$, so the monic polynomial $x-y^2$ is adjoined. The new $S$-polynomials give $y^3-1$: indeed $S(xy-1, x-y^2) = (xy-1) - y(x-y^2) = y^3-1$, whose leading monomial $y^3$ is divisible by no earlier one, so $y^3-1$ is adjoined. All remaining $S$-polynomials reduce to $0$ modulo $G = \{xy-1,\ x^2-y,\ x-y^2,\ y^3-1\}$; reducing this set gives the reduced Gröbner basis

$$ G = \{x - y^2,\ y^3 - 1\}, $$

since $xy - 1 - y(x-y^2) = y^3-1$ and $x^2 - y - x(x-y^2) = y(xy-1)$, both reductions landing in $\langle G\rangle$. The common zeros are then read off directly: $y^3 = 1$ and $x = y^2$, so the three points $(1,1)$, $(\omega,\omega^2)$ and $(\omega^2,\omega)$ over a field containing a primitive cube root of unity $\omega$.

Elimination Theory

The Elimination Theorem

Definition. Given an ideal $I \subseteq K[x_1,\ldots,x_n]$ and $0 \leq k \leq n$, the $k$-th elimination ideal is

$$ I_k = I \cap K[x_{k+1}, \ldots, x_n], $$

the set of polynomials of $I$ involving none of $x_1,\ldots,x_k$; it is an ideal of the smaller polynomial ring.

Theorem (elimination theorem). Fix the lexicographic order with $x_1 \succ x_2 \succ \cdots \succ x_n$ and let $G$ be a Gröbner basis of $I$ with respect to it. Then for each $k$ the set

$$ G_k = G \cap K[x_{k+1},\ldots,x_n] $$

is a Gröbner basis of the $k$-th elimination ideal $I_k$.

Proof. Let $f \in I_k$ be nonzero. By the Gröbner property some $\operatorname{LM}(g)$ with $g \in G$ divides $\operatorname{LM}(f)$. Since $f$ involves none of $x_1,\ldots,x_k$, its leading monomial does not involve them either, so neither does $\operatorname{LM}(g)$, and by the lexicographic order a monomial involving some $x_i$ with $i \leq k$ is larger than any monomial of the same total degree in the remaining variables; hence $g$ involves none of $x_1,\ldots,x_k$ and lies in $G_k$. Thus $\operatorname{LT}(I_k) \subseteq \langle \operatorname{LT}(g) : g \in G_k\rangle$, and the reverse inclusion is clear.

Example. For $I = \langle xy-1,\ x^2-y \rangle$ with lex $x \succ y$ and $G = \{x-y^2,\ y^3-1\}$: the first elimination ideal is $I_1 = I \cap K[y] = \langle y^3-1\rangle$, so the projection of the variety onto the $y$-axis is the set of cube roots of unity, and for each of them the equation $x = y^2$ determines $x$ uniquely. This is the shape that makes a polynomial system solvable: the elimination ideal gives a univariate polynomial whose roots are the values of the last variable, and the remaining equations are triangular.

Theorem (implicitisation). Let the curve in $\mathbb{A}^3$ be parametrised by $x = t$, $y = t^2$, $z = t^3$, that is, let $I = \langle x-t,\ y-t^2,\ z-t^3\rangle \subseteq K[x,y,z,t]$ with lex $x \succ y \succ z \succ t$. Then

$$ I \cap K[x,y,z] = \langle y - x^2,\ z - x^3 \rangle , $$

so the implicit equations of the twisted cubic are $y = x^2$, $z = x^3$; every point of the form $(t,t^2,t^3)$ satisfies them, and conversely every solution of the two equations is of that form over an algebraically closed field.

Proof sketch. The polynomial $y - x^2 = (y - t^2) - (x^2-t^2)$ lies in $I$ because $x^2 - t^2 = (x-t)(x+t)$, and similarly $z - x^3 \in I$; the elimination computation shows that these two generate the elimination ideal, and the converse statement is the Extension theorem below, which applies because the leading coefficients in the elimination variables are constants.

The Extension Theorem and the Nullstellensatz

Theorem (Extension theorem). Fix lex $x_1 \succ \cdots \succ x_n$ and let $I \subseteq K[x_1,\ldots,x_n]$, $I_1 = I \cap K[x_2,\ldots,x_n]$. Let $G_1$ be a Gröbner basis of $I_1$ and let $g_1,\ldots,g_m$ be those elements of the Gröbner basis $G$ of $I$ whose leading monomials do not involve $x_1$, written as polynomials in $x_1$ with coefficients in $K[x_2,\ldots,x_n]$. If $(a_2,\ldots,a_n)$ is a common zero of $I_1$ at which none of $\operatorname{LC}_{x_1}(g_1),\ldots,\operatorname{LC}_{x_1}(g_m)$ vanishes and if $K$ is algebraically closed, then there is $a_1 \in K$ with $(a_1,\ldots,a_n)$ a common zero of $I$.

Proof sketch. Substitute $(a_2,\ldots,a_n)$: the polynomials $g_i$ become univariate polynomials in $x_1$ of degrees equal to their $x_1$-degrees, the highest of which has a nonzero leading coefficient by hypothesis, so the resulting univariate ideal is proper and has a root in the algebraically closed field $K$; that root is $a_1$. The failure of the hypothesis is exactly the phenomenon of a partial solution not extending, such as the system $x_1x_2 = 1$, $x_1x_3 = 1$ at the partial solution $x_2 = x_3 = 0$.

Theorem (Nullstellensatz, effective form). Let $I \subseteq K[x_1,\ldots,x_n]$ and $f \in K[x_1,\ldots,x_n]$, and suppose $f$ vanishes at every common zero of $I$ in $\bar K^n$. Then $f$ lies in the radical of $I$, and this can be decided by a Gröbner computation: with $t$ a new variable, $f \in \sqrt I$ if and only if

$$ 1 \in \langle I,\ 1 - tf \rangle \subseteq K[x_1,\ldots,x_n,t] , $$

which is tested by computing a Gröbner basis of the enlarged ideal and reducing $1$ by it.

Proof sketch. If $f \in \sqrt I$ then $f^m \in I$ for some $m$ and $1 = t^mf^m + (1-t^mf^m) \in \langle I, 1-tf\rangle$, since $1 - t^mf^m$ is divisible by $1-tf$. Conversely, if $1 = \sum a_ig_i + a(1-tf)$ with $g_i \in I$, substitute $t = 1/f$ in the identity multiplied by a high power of $f$ to clear denominators; the resulting identity exhibits a power of $f$ as an element of $I$. This is the Rabinowitsch trick.

Theorem (ideal quotient and saturation). Let $I, J \subseteq R$ with $J = \langle h_1,\ldots,h_r\rangle$. Then

$$ I : J = \bigcap_{i=1}^{r}\bigl(I : h_i\bigr), \qquad I : h = \frac{1}{h}\bigl(I \cap \langle h\rangle\bigr), $$

and $I : h$ is computed by introducing a new variable $t$, forming $I + \langle 1 - th\rangle$ in $R[t]$, and taking the elimination ideal $K[x_1,\ldots,x_n]$. The saturation $I : h^\infty = \bigcup_m (I : h^m)$ agrees with $I : h$ for $m$ large enough, the union stabilising because $R$ is Noetherian, and it is computed as the elimination ideal of $I + \langle 1-th\rangle$ after the removal of the $h$-primary components.

Example. The projective closure of a curve is computed by the same device. The affine twisted cubic $\{(t,t^2,t^3)\}$ has, in the projective coordinates $(x:y:z:w)$, the parametrisation $(1:t:t^2:t^3)$, and its homogeneous ideal is generated by the three $2\times2$ minors of the matrix

$$ \begin{pmatrix} x & y & z\\ y & z & w\end{pmatrix}, $$

namely $xz-y^2$, $xw-yz$ and $yw-z^2$; setting $w = 1$ recovers the two affine equations $y = x^2$, $z = x^3$ up to the chart condition, and the third minor becomes trivial. The passage from the affine elimination ideal to the homogeneous ideal is the homogenisation of the elimination computation, a standard application of the same algorithm.

Applications

Solving Systems and the Shape Lemma

Definition. An ideal $I \subseteq K[x_1,\ldots,x_n]$ is zero-dimensional if the quotient ring $K[x_1,\ldots,x_n]/I$ is a finite-dimensional $K$-vector space; equivalently, if $I$ contains a polynomial in each variable alone. The dimension of the quotient is the number of points of the variety counted with multiplicity over an algebraically closed field, when the variety is finite.

Theorem (shape lemma). Let $I$ be a zero-dimensional radical ideal in general position with respect to $x_n$ — that is, the finitely many points of its variety have distinct $x_n$-coordinates. Then the reduced lexicographic Gröbner basis has the shape

$$ G = \{x_1 - g_1(x_n),\ \ldots,\ x_{n-1}-g_{n-1}(x_n),\ g(x_n)\}, $$

with $\deg g$ equal to the number of points; the roots of $g$ are the $x_n$-coordinates of the points and the other coordinates are obtained by evaluating the $g_i$.

Proof sketch. Zero-dimensionality gives $I_k = \langle g\rangle$ for the last elimination ideal, with $\deg g$ the number of points by the Extension theorem; radicality plus general position makes the map from the points to the $x_n$-line injective, so Lagrange interpolation produces, for each $i < n$, the polynomial $g_i$ of degree less than $\deg g$ taking the value of $x_i$ at each point, and $x_i - g_i(x_n)$ lies in $I$ by the effective Nullstellensatz applied to the finite variety.

Example. For $I = \langle x^2+y^2-1,\ x-y\rangle \subseteq \mathbb{R}[x,y]$ with lex $x \succ y$: the reduced Gröbner basis is $\{x-y,\ y^2-\frac12\}$, of the shape above with $g(y) = y^2-\frac12$ and $x = y$; the solutions are $y = \pm\sqrt2/2$ with $x = y$, the two points in which the line $x = y$ meets the unit circle.

Elimination and Curves

Theorem (intersection of plane curves). Let $f, g \in K[x,y]$ have no common factor, with degrees $m$ and $n$. Then the elimination ideal $\langle f, g\rangle \cap K[y]$ is nonzero and is generated by a polynomial of degree $mn$; its roots are the $y$-coordinates of the points of $f = g = 0$, and the multiplicity of a root is the sum of the intersection multiplicities of the points with that $y$-coordinate. This is the resultant $R_y(f,g) = \operatorname{Res}_y(f,g)$, a polynomial of degree $mn$ computed either as the determinant of the Sylvester matrix or by an elimination computation, and it is the effective form of Bézout's theorem as stated in Algebraic Curves.

Example. For $f = x^2+y^2-1$ and $g = x-y$: the resultant in $x$ gives $2y^2-1$, of degree $2 = 1\cdot2$, with the two simple roots found above. For $f = x^2+y^2-1$ and $g = x^2-y$, both of degree $2$: eliminating $x$ by subtracting the two equations gives $y^2+y-1 = 0$, and since the two curves are symmetric in $x$ the resultant in $x$ is $(y^2+y-1)^2$, of degree $4 = 2\cdot2$; its roots are the $y$-coordinates of the four intersection points over $\mathbb{C}$, namely $y = (-1\pm\sqrt5)/2$ with $x^2 = y$, two of which have real coordinates.

Example (an ideal-theoretic computation). The intersection of the two ideals $I = \langle x^2, xy\rangle$ and $J = \langle y\rangle$ is $I \cap J = \langle x^2y, xy\rangle = \langle xy\rangle$, and this is computed by the elimination $I \cap J = (tI + (1-t)J)\cap R$ in $R[t]$; the notion of an intersection of ideals and the associated primary decomposition are those of Primary Decomposition.

Summary

A monomial order is a well-ordering of the monomials compatible with multiplication; Dickson's lemma, that every monomial ideal is finitely generated, gives the Hilbert basis theorem and the termination of all the algorithms. Division of a polynomial by an ordered tuple of polynomials leaves a remainder that depends on the order, and a Gröbner basis is a generating set for which the remainder is unique and vanishes exactly on the ideal; every ideal of $K[x_1,\ldots,x_n]$ has a Gröbner basis with respect to every monomial order and a unique reduced one, and Buchberger's algorithm computes it in finitely many steps from any finite generating set, testing the termination by the criterion that all $S$-polynomials reduce to zero.

With a Gröbner basis in h, membership and equality of ideals are decided, the elimination ideals $I \cap K[x_{k+1},\ldots,x_n]$ are read directly off a lexicographic Gröbner basis, the implicit equations of a parametrised variety are obtained by elimination, the effective Nullstellensatz decides membership in a radical by the test $1 \in \langle I, 1-tf\rangle$, ideal quotients and saturations are computed by the same device, and a zero-dimensional ideal in general position is triangularised by the shape lemma, so that a polynomial system is solved by finding the roots of one univariate polynomial and back-substituting. The resultant is the classical form of the elimination in two variables and gives the degree $mn$ of Bézout's theorem; the projective and geometric reading of these computations belongs to Algebraic Curves, and the deeper invariants of the theory — Hilbert functions, resolutions, syzygies — to the Part II agent's.

Summary of Notation

Symbol	Meaning
$K$	Field of coefficients
$R = K[x_1,\ldots,x_n]$	Polynomial ring in $n$ variables
$x^\alpha$	Monomial with exponent vector $\alpha$
$\succ$, $\succ_{\mathrm{lex}}$, $\succ_{\mathrm{grlex}}$, $\succ_{\mathrm{grevlex}}$	A monomial order and the three standard ones
$\operatorname{LT}(f)$, $\operatorname{LM}(f)$, $\operatorname{LC}(f)$	Leading term, monomial, coefficient
$\operatorname{LT}(I)$	Ideal of leading terms
$G$	A Gröbner basis
$S(f,g)$	The $S$-polynomial
$I_k = I \cap K[x_{k+1},\ldots,x_n]$	The $k$-th elimination ideal
$I : J$, $I : h^\infty$	Ideal quotient, saturation
$\sqrt I$	Radical of $I$
$\operatorname{Res}_y(f,g)$	Resultant in $y$
$g(x_n)$, $g_i(x_n)$	Univariate polynomials of the shape lemma