main theorem of elimination theory

The affine plane over a field {{mvar|k}} is the direct product $A\_2=L\_x\backslash times\; L\_y$ of two copies of {{mvar|k}}. Let

:$\backslash pi\backslash colon\; L\_x\backslash times\; L\_y\; \backslash to\; L\_x$

be the projection

:$(x,y)\backslash mapsto\; \backslash pi(x,y)=x.$

This projection is not closed for the Zariski topology (nor for the usual topology if $k=\; \backslash R$ or $k=\; \backslash C$), because the image by $\backslash pi$ of

the hyperbola {{mvar|H}} of equation $xy-1=0$ is $L\_x\backslash setminus\; \backslash \{0\backslash \},$ which is not closed, although {{mvar|H}} 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

:$xy\_1-y\_0=0,$

and contains

:$\backslash overline\backslash pi(0,(1,0))=0,$

where $\backslash overline\backslash pi$ is the prolongation of $\backslash pi$ to $L\_x\backslash times\; P\_y.$

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 {{mvar|y}}-axis.

More generally, the image by $\backslash pi$ of every algebraic set in $L\_x\backslash times\; L\_y$ is either a finite number of points, or $L\_x$ with a finite number of points removed, while the image by $\backslash overline\backslash pi$ of any algebraic set in $L\_x\backslash times\; P\_y$ is either a finite number of points or the whole line $L\_y.$ It follows that the image by $\backslash overline\backslash pi$ of any algebraic set is an algebraic set, that is that $\backslash overline\backslash pi$ 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 $R[\backslash mathbf\; x]=R[x\_1,\; \backslash ldots,\; x\_n]$ over a commutative Noetherian ring {{mvar|R}}, and a homogeneous ideal {{mvar|I}} generated by homogeneous polynomials $f\_1,\backslash ldots,\; f\_k.$ (In the original proof by Macaulay, {{mvar|k}} was equal to {{mvar|n}}, and {{mvar|R}} was a polynomial ring over the integers, whose indeterminates were all the coefficients of the$f\_i\backslash mathrm\; s.$)

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

The theorem is: there is an ideal $\backslash mathfrak\; r$ in {{mvar|R}}, uniquely determined by {{mvar|I}}, such that, for every ring homomorphism $\backslash varphi$ from {{mvar|R}} into a field {{mvar|K}}, the homogeneous polynomials $\backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_k)$ have a nontrivial common zero (in an algebraic closure of {{mvar|K}}) if and only if $\backslash varphi(\backslash mathfrak\; r)=\backslash \{0\backslash \}.$

Moreover, $\backslash mathfrak\; r\; =0$ if {{math|1=k < n}}, and $\backslash mathfrak\; r$ is principal if {{math|1=k = n}}. In this latter case, a generator of $\backslash mathfrak\; r$ is called the resultant of $f\_1,\backslash ldots,\; f\_k.$

Using above notation, one has first to characterize the condition that $\backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_k)$ do not have any non-trivial common zero. This is the case if the maximal homogeneous ideal $\backslash mathfrak\; m\; =\; \backslash langle\; x\_1,\; \backslash ldots,\; x\_n\backslash rangle$ is the only homogeneous prime ideal containing $\backslash varphi(I)=\backslash langle\; \backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_k)\backslash rangle.$ Hilbert's Nullstellensatz asserts that this is the case if and only if $\backslash varphi(I)$ contains a power of each $x\_i,$ or, equivalently, that $\backslash mathfrak\; m^d\; \backslash subseteq\; \backslash varphi(I)$ for some positive integer {{mvar|d}}.

For this study, Macaulay introduced a matrix that is now called Macaulay matrix in degree {{mvar|d}}. Its rows are indexed by the monomials of degree {{mvar|d}} in $x\_1,\; \backslash ldots,\; x\_n,$ and its columns are the vectors of the coefficients on the monomial basis of the polynomials of the form $m\backslash varphi(f\_i),$ where {{mvar|m}} is a monomial of degree $d-\backslash deg(f\_i).$ One has $\backslash mathfrak\; m^d\; \backslash subseteq\; \backslash varphi(I)$ if and only if the rank of the Macaulay matrix equals the number of its rows.

If {{math|k < n}}, the rank of the Macaulay matrix is lower than the number of its rows for every {{mvar|d}}, and, therefore, $\backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_k)$ 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 $d\_2\backslash ge\; d\_3\; \backslash ge\backslash cdots\backslash ge\; d\_k\backslash ge\; d\_1.$ The degree

:$D=\; d\_1+d\_2+\backslash cdots+d\_n-n+1\; =\; 1+\backslash sum\_\{i=1\}^n\; (d\_i-1)$

is called Macaulay's degree or Macaulay's bound because Macaulay's has proved that $\backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_k)$ have a non-trivial common zero if and only if the rank of the Macaulay matrix in degree {{mvar|D}} is lower than the number to its rows. In other words, the above {{mvar|d}} may be chosen once for all as equal to {{mvar|D}}.

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

If {{math|1=k = n}}, Macaulay has also proved that $\backslash mathfrak\; r$ is a principal ideal (although Macaulay matrix in degree {{mvar|D}} is not a square matrix when {{math|k > 2}}), which is generated by the resultant of $\backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_n).$ This ideal is also generically a prime ideal, as it is prime if {{mvar|R}} is the ring of integer polynomials with the all coefficients of $\backslash varphi(f\_1),\backslash ldots,\; \backslash varphi(f\_k)$ as indeterminates.

In the preceding formulation, the polynomial ring $R[\backslash mathbf\; x]=R[x\_1,\; \backslash ldots,\; x\_n]$ defines a morphism of schemes (which are algebraic varieties if {{mvar|R}} is finitely generated over a field)

:$\backslash mathbb\{P\}^\{n-1\}\_R\; =\; \backslash operatorname\{Proj\}(R[\backslash mathbf\; x])\; \backslash to\; \backslash operatorname\{Spec\}(R).$

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

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

• {{cite book|last=Mumford|first=David|title=The Red Book of Varieties and Schemes|publisher=Springer|year=1999|isbn=9783540632931|author-link=David Mumford}}
• {{cite book|last=Eisenbud|first=David|title=Commutative Algebra: with a View Toward Algebraic Geometry|publisher=Springer|year=2013|isbn=9781461253501|author-link=David Eisenbud}}
• {{cite book|last=Milne|first=James S.|title=The Abel Prize 2008–2012|chapter=The Work of John Tate|publisher=Springer|year=2014|isbn=9783642394492|author-link=James Milne (mathematician)}}

Category:Theorems in algebraic geometry