Main theorem of elimination theory

Template:Short description In algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following more classical setting. Let Template:Math be a field, denote by ${\mathbb{P}}_k^n$ the Template:Math-dimensional projective space over Template:Math. The main theorem of elimination theory is the statement that for any Template:Math and any algebraic variety Template:Mvar defined over Template:Math, the projection map $V\times {\mathbb{P}}_k^n\to V$ sends Zariski-closed subsets to Zariski-closed subsets.

The main theorem of elimination theory is a corollary and a generalization of Macaulay's theory of multivariate resultant. The resultant of Template:Mvar homogeneous polynomials in Template:Mvar variables is the value of a polynomial function of the coefficients, which takes the value zero if and only if the polynomials have a common non-trivial zero over some field containing the coefficients.

This belongs to elimination theory, as computing the resultant amounts to eliminate variables between polynomial equations. In fact, given a system of polynomial equations, which is homogeneous in some variables, the resultant eliminates these homogeneous variables by providing an equation in the other variables, which has, as solutions, the values of these other variables in the solutions of the original system.

A simple motivating example

The affine plane over a field Template:Mvar is the direct product $A_2=L_x\times L_y$ of two copies of Template:Mvar. Let

$\pi :L_x\times L_y\to L_x$

be the projection

$(x,y)\mapsto \pi (x,y)=x.$

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

If one extends $L_y$ to a projective line $P_y,$ the equation of the projective completion of the hyperbola becomes

$x{y}_1-y_0=0,$

and contains

$\bar{\pi }(0,(1,0))=0,$

where $\bar{\pi }$ is the prolongation of $\pi$ to $L_x\times P_y.$

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 Template:Mvar-axis.

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

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

Classical formulation

For stating the theorem in terms of commutative algebra, one has to consider a polynomial ring $R[𝐱]=R[x_1,\dots ,x_n]$ over a commutative Noetherian ring Template:Mvar, and a homogeneous ideal Template:Mvar generated by homogeneous polynomials $f_1,\dots ,f_k.$ (In the original proof by Macaulay, Template:Mvar was equal to Template:Mvar, and Template:Mvar was a polynomial ring over the integers, whose indeterminates were all the coefficients of the$f_i\mathrm{s}.$)

Any ring homomorphism $\phi$ from Template:Mvar into a field Template:Mvar, defines a ring homomorphism $R[𝐱]\to K[𝐱]$ (also denoted $\phi$), by applying $\phi$ to the coefficients of the polynomials.

The theorem is: there is an ideal $𝔯$ in Template:Mvar, uniquely determined by Template:Mvar, such that, for every ring homomorphism $\phi$ from Template:Mvar into a field Template:Mvar, the homogeneous polynomials $\phi (f_1),\dots ,\phi (f_k)$ have a nontrivial common zero (in an algebraic closure of Template:Mvar) if and only if $\phi (𝔯)=\{0\}.$

Moreover, $𝔯=0$ if Template:Math, and $𝔯$ is principal if Template:Math. In this latter case, a generator of $𝔯$ is called the resultant of $f_1,\dots ,f_k.$

Hints for a proof and related results

Using above notation, one has first to characterize the condition that $\phi (f_1),\dots ,\phi (f_k)$ do not have any non-trivial common zero. This is the case if the maximal homogeneous ideal $𝔪=⟨x_1,\dots ,x_n⟩$ is the only homogeneous prime ideal containing $\phi (I)=⟨\phi (f_1),\dots ,\phi (f_k)⟩.$ Hilbert's Nullstellensatz asserts that this is the case if and only if $\phi (I)$ contains a power of each $x_i,$ or, equivalently, that ${𝔪}^d\subseteq \phi (I)$ for some positive integer Template:Mvar.

For this study, Macaulay introduced a matrix that is now called Macaulay matrix in degree Template:Mvar. Its rows are indexed by the monomials of degree Template:Mvar in $x_1,\dots ,x_n,$ and its columns are the vectors of the coefficients on the monomial basis of the polynomials of the form $m\phi (f_i),$ where Template:Mvar is a monomial of degree $d-\mathrm{deg}(f_i).$ One has ${𝔪}^d\subseteq \phi (I)$ if and only if the rank of the Macaulay matrix equals the number of its rows.

If Template:Math, the rank of the Macaulay matrix is lower than the number of its rows for every Template:Mvar, and, therefore, $\phi (f_1),\dots ,\phi (f_k)$ have always a non-trivial common zero.

Otherwise, let $d_i$ be the degree of $f_i,$ and suppose that the indices are chosen in order that $d_2\ge d_3\ge \cdots \ge d_k\ge d_1.$ The degree

$D=d_1+d_2+\cdots +d_n-n+1=1+\sum _{i=1}^n(d_i-1)$

is called Macaulay's degree or Macaulay's bound because Macaulay's has proved that $\phi (f_1),\dots ,\phi (f_k)$ have a non-trivial common zero if and only if the rank of the Macaulay matrix in degree Template:Mvar is lower than the number to its rows. In other words, the above Template:Mvar may be chosen once for all as equal to Template:Mvar.

Therefore, the ideal $𝔯,$ whose existence is asserted by the main theorem of elimination theory, is the zero ideal if Template:Math, and, otherwise, is generated by the maximal minors of the Macaulay matrix in degree Template:Mvar.

If Template:Math, Macaulay has also proved that $𝔯$ is a principal ideal (although Macaulay matrix in degree Template:Mvar is not a square matrix when Template:Math), which is generated by the resultant of $\phi (f_1),\dots ,\phi (f_n).$ This ideal is also generically a prime ideal, as it is prime if Template:Mvar is the ring of integer polynomials with the all coefficients of $\phi (f_1),\dots ,\phi (f_k)$ as indeterminates.

Geometrical interpretation

In the preceding formulation, the polynomial ring $R[𝐱]=R[x_1,\dots ,x_n]$ defines a morphism of schemes (which are algebraic varieties if Template:Mvar is finitely generated over a field)

${\mathbb{P}}_R^{n-1}=\mathrm{Proj}(R[𝐱])\to \mathrm{Spec}(R).$

The theorem asserts that the image of the Zariski-closed set Template:Math defined by Template:Mvar is the closed set Template:Math. Thus the morphism is closed.

See also

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

References

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