Main theorem of elimination theory

The affine plane over a field k is the direct product ${\displaystyle A_{2}=L_{x}\times L_{y}}$ of two copies of k. Let

${\displaystyle \pi \colon L_{x}\times L_{y}\to L_{x}}$

be the projection

${\displaystyle (x,y)\mapsto \pi (x,y)=x.}$

This projection is not closed for the Zariski topology (nor for the usual topology if ${\displaystyle k=\mathbb {R} }$ or ${\displaystyle k=\mathbb {C} }$), because the image by ${\displaystyle \pi }$ of the hyperbola H of equation ${\displaystyle xy-1=0}$ is ${\displaystyle L_{x}\setminus \{0\},}$ which is not closed, although H is closed, being an algebraic variety.

If one extends ${\displaystyle L_{y}}$ to a projective line ${\displaystyle P_{y},}$ the equation of the projective completion of the hyperbola becomes

${\displaystyle xy_{1}-y_{0}=0,}$

and contains

${\displaystyle {\overline {\pi }}(0,(1,0))=0,}$

where ${\displaystyle {\overline {\pi }}}$ is the prolongation of ${\displaystyle \pi }$ to ${\displaystyle L_{x}\times P_{y}.}$

This is commonly expressed by saying the origin of the affine plane is the projection of the point of the hyperbola that is at infinity, in the direction of the y-axis.

More generally, the image by ${\displaystyle \pi }$ of every algebraic set in ${\displaystyle L_{x}\times L_{y}}$ is either a finite number of points, or ${\displaystyle L_{x}}$ with a finite number of points removed, while the image by ${\displaystyle {\overline {\pi }}}$ of any algebraic set in ${\displaystyle L_{x}\times P_{y}}$ is either a finite number of points or the whole line ${\displaystyle L_{y}.}$ It follows that the image by ${\displaystyle {\overline {\pi }}}$ of any algebraic set is an algebraic set, that is that ${\displaystyle {\overline {\pi }}}$ is a closed map for Zariski topology.

The main theorem of elimination theory is a wide generalization of this property.

For stating the theorem in terms of commutative algebra, one has to consider a polynomial ring ${\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]}$ over a commutative Noetherian ring R, and a homogeneous ideal I generated by homogeneous polynomials ${\displaystyle f_{1},\ldots ,f_{k}.}$ (In the original proof by Macaulay, k was equal to n, and R was a polynomial ring over the integers, whose indeterminates were all the coefficients of the${\displaystyle f_{i}\mathrm {s} .}$)

Any ring homomorphism ${\displaystyle \varphi }$ from R into a field K, defines a ring homomorphism ${\displaystyle R[\mathbf {x} ]\to K[\mathbf {x} ]}$ (also denoted ${\displaystyle \varphi }$), by applying ${\displaystyle \varphi }$ to the coefficients of the polynomials.

The theorem is: there is an ideal ${\displaystyle {\mathfrak {r}}}$ in R, uniquely determined by I, such that, for every ring homomorphism ${\displaystyle \varphi }$ from R into a field K, the homogeneous polynomials ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}$ have a nontrivial common zero (in an algebraic closure of K) if and only if ${\displaystyle \varphi ({\mathfrak {r}})=\{0\}.}$

Moreover, ${\displaystyle {\mathfrak {r}}=0}$ if k < n, and ${\displaystyle {\mathfrak {r}}}$ is principal if k = n. In this latter case, a generator of ${\displaystyle {\mathfrak {r}}}$ is called the resultant of ${\displaystyle f_{1},\ldots ,f_{k}.}$

Using above notation, one has first to characterize the condition that ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}$ do not have any non-trivial common zero. This is the case if the maximal homogeneous ideal ${\displaystyle {\mathfrak {m}}=\langle x_{1},\ldots ,x_{n}\rangle }$ is the only homogeneous prime ideal containing ${\displaystyle \varphi (I)=\langle \varphi (f_{1}),\ldots ,\varphi (f_{k})\rangle .}$ Hilbert's Nullstellensatz asserts that this is the case if and only if ${\displaystyle \varphi (I)}$ contains a power of each ${\displaystyle x_{i},}$ or, equivalently, that ${\displaystyle {\mathfrak {m}}^{d}\subseteq \varphi (I)}$ for some positive integer d.

For this study, Macaulay introduced a matrix that is now called Macaulay matrix in degree d. Its rows are indexed by the monomials of degree d in ${\displaystyle x_{1},\ldots ,x_{n},}$ and its columns are the vectors of the coefficients on the monomial basis of the polynomials of the form ${\displaystyle m\varphi (f_{i}),}$ where m is a monomial of degree ${\displaystyle d-\deg(f_{i}).}$ One has ${\displaystyle {\mathfrak {m}}^{d}\subseteq \varphi (I)}$ if and only if the rank of the Macaulay matrix equals the number of its rows.

If k < n, the rank of the Macaulay matrix is lower than the number of its rows for every d, and, therefore, ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}$ have always a non-trivial common zero.

Otherwise, let ${\displaystyle d_{i}}$ be the degree of ${\displaystyle f_{i},}$ and suppose that the indices are chosen in order that ${\displaystyle d_{2}\geq d_{3}\geq \cdots \geq d_{k}\geq d_{1}.}$ The degree

${\displaystyle D=d_{1}+d_{2}+\cdots +d_{n}-n+1=1+\sum _{i=1}^{n}(d_{i}-1)}$

is called Macaulay's degree or Macaulay's bound because Macaulay's has proved that ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}$ have a non-trivial common zero if and only if the rank of the Macaulay matrix in degree D is lower than the number to its rows. In other words, the above d may be chosen once for all as equal to D.

Therefore, the ideal ${\displaystyle {\mathfrak {r}},}$ whose existence is asserted by the main theorem of elimination theory, is the zero ideal if k < n, and, otherwise, is generated by the maximal minors of the Macaulay matrix in degree D.

If k = n, Macaulay has also proved that ${\displaystyle {\mathfrak {r}}}$ is a principal ideal (although Macaulay matrix in degree D is not a square matrix when k > 2), which is generated by the resultant of ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{n}).}$ This ideal is also generically a prime ideal, as it is prime if R is the ring of integer polynomials with the all coefficients of ${\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}$ as indeterminates.

In the preceding formulation, the polynomial ring ${\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]}$ defines a morphism of schemes (which are algebraic varieties if R is finitely generated over a field)

${\displaystyle \mathbb {P} _{R}^{n-1}=\operatorname {Proj} (R[\mathbf {x} ])\to \operatorname {Spec} (R).}$

The theorem asserts that the image of the Zariski-closed set V(I) defined by I is the closed set V(r). Thus the morphism is closed.

• Elimination of quantifiers
• Macaulay's resultant
• Gröbner basis

• Mumford, David (1999). The Red Book of Varieties and Schemes. Springer. ISBN 9783540632931.
• Eisenbud, David (2013). Commutative Algebra: with a View Toward Algebraic Geometry. Springer. ISBN 9781461253501.
• Milne, James S. (2014). "The Work of John Tate". The Abel Prize 2008–2012. Springer. ISBN 9783642394492.