Main theorem of elimination theory

<h2 id="a-simple-motivating-example">A simple motivating example</h2>
The <a href="/facts/Affine_plane/tPGR5SMQ">affine plane</a> over a field k is the <a href="/facts/Direct_product/GehGQmNV">direct product</a> 
 
 
 
 
 A
 
 2
 
 
 =
 
 L
 
 x
 
 
 ×
 
 L
 
 y
 
 
 
 
 {\displaystyle A_{2}=L_{x}\times L_{y}}
 
 of two copies of k. Let

π
        :
        
          L
          
            x
          
        
        ×
        
          L
          
            y
          
        
        →
        
          L
          
            x
          
        
      
    
    {\displaystyle \pi \colon L_{x}\times L_{y}\to L_{x}}

be the projection

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

This projection is not <a href="/facts/Closed_map/abHUKX7l">closed</a> for the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a> (nor for the usual topology if 
 
 
 
 k
 =
 
 R
 
 
 
 {\displaystyle k=\mathbb {R} }
 
 or 
 
 
 
 k
 =
 
 C
 
 
 
 {\displaystyle k=\mathbb {C} }
 
), because the image by 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 of
the <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> H of equation 
 
 
 
 x
 y
 −
 1
 =
 0
 
 
 {\displaystyle xy-1=0}
 
 is 
 
 
 
 
 L
 
 x
 
 
 ∖
 {
 0
 }
 ,
 
 
 {\displaystyle L_{x}\setminus \{0\},}
 
 which is not closed, although H is closed, being an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a>.
If one extends 
 
 
 
 
 L
 
 y
 
 
 
 
 {\displaystyle L_{y}}
 
 to a projective line 
 
 
 
 
 P
 
 y
 
 
 ,
 
 
 {\displaystyle P_{y},}
 
 the equation of the <a href="/facts/Projective_completion/aB2XdB7B">projective completion</a> of the hyperbola becomes

x
        
          y
          
            1
          
        
        −
        
          y
          
            0
          
        
        =
        0
        ,
      
    
    {\displaystyle xy_{1}-y_{0}=0,}

and contains

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

where 
 
 
 
 
 
 π
 ¯
 
 
 
 
 {\displaystyle {\overline {\pi }}}
 
 is the prolongation of 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 to 
 
 
 
 
 L
 
 x
 
 
 ×
 
 P
 
 y
 
 
 .
 
 
 {\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 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 of every algebraic set in 
 
 
 
 
 L
 
 x
 
 
 ×
 
 L
 
 y
 
 
 
 
 {\displaystyle L_{x}\times L_{y}}
 
 is either a finite number of points, or 
 
 
 
 
 L
 
 x
 
 
 
 
 {\displaystyle L_{x}}
 
 with a finite number of points removed, while the image by 
 
 
 
 
 
 π
 ¯
 
 
 
 
 {\displaystyle {\overline {\pi }}}
 
 of any algebraic set in 
 
 
 
 
 L
 
 x
 
 
 ×
 
 P
 
 y
 
 
 
 
 {\displaystyle L_{x}\times P_{y}}
 
 is either a finite number of points or the whole line 
 
 
 
 
 L
 
 y
 
 
 .
 
 
 {\displaystyle L_{y}.}
 
 It follows that the image by 
 
 
 
 
 
 π
 ¯
 
 
 
 
 {\displaystyle {\overline {\pi }}}
 
 of any algebraic set is an algebraic set, that is that 
 
 
 
 
 
 π
 ¯
 
 
 
 
 {\displaystyle {\overline {\pi }}}
 
 is a closed map for Zariski topology.
The main theorem of elimination theory is a wide generalization of this property.

<h2 id="classical-formulation">Classical formulation</h2>
For stating the theorem in terms of <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>, one has to consider a <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> 
 
 
 
 R
 [
 
 x
 
 ]
 =
 R
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 
 
 {\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]}
 
 over a commutative <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> R, and a <a href="/facts/Homogeneous_ideal/AIUjkSaP">homogeneous ideal</a> I generated by <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomials</a> 
 
 
 
 
 f
 
 1
 
 
 ,
 …
 ,
 
 f
 
 k
 
 
 .
 
 
 {\displaystyle f_{1},\ldots ,f_{k}.}
 
 (In the original proof by <a href="/facts/Francis_Sowerby_Macaulay/4XTyDBNa">Macaulay</a>, k was equal to n, and R was a polynomial ring over the integers, whose indeterminates were all the coefficients of the
 
 
 
 
 f
 
 i
 
 
 
 s
 
 .
 
 
 {\displaystyle f_{i}\mathrm {s} .}
 
)
Any <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 from R into a field K, defines a ring homomorphism 
 
 
 
 R
 [
 
 x
 
 ]
 →
 K
 [
 
 x
 
 ]
 
 
 {\displaystyle R[\mathbf {x} ]\to K[\mathbf {x} ]}
 
 (also denoted 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
), by applying 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 to the coefficients of the polynomials.
The theorem is: there is an ideal 
 
 
 
 
 
 r
 
 
 
 
 {\displaystyle {\mathfrak {r}}}
 
 in R, uniquely determined by I, such that, for every ring homomorphism 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 from R into a field K, the homogeneous polynomials 
 
 
 
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 k
 
 
 )
 
 
 {\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}
 
 have a nontrivial common zero (in an algebraic closure of K) if and only if 
 
 
 
 φ
 (
 
 
 r
 
 
 )
 =
 {
 0
 }
 .
 
 
 {\displaystyle \varphi ({\mathfrak {r}})=\{0\}.}

Moreover, 
 
 
 
 
 
 r
 
 
 =
 0
 
 
 {\displaystyle {\mathfrak {r}}=0}
 
 if k < n, and 
 
 
 
 
 
 r
 
 
 
 
 {\displaystyle {\mathfrak {r}}}
 
 is <a href="/facts/Principal_ideal/Cga66qgM">principal</a> if k = n. In this latter case, a generator of 
 
 
 
 
 
 r
 
 
 
 
 {\displaystyle {\mathfrak {r}}}
 
 is called the <a href="/facts/Macaulay%27s_resultant/A5g2MdhT">resultant</a> of 
 
 
 
 
 f
 
 1
 
 
 ,
 …
 ,
 
 f
 
 k
 
 
 .
 
 
 {\displaystyle f_{1},\ldots ,f_{k}.}

<h2 id="hints-for-a-proof-and-related-results">Hints for a proof and related results</h2>
Using above notation, one has first to characterize the condition that 
 
 
 
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 k
 
 
 )
 
 
 {\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 
 
 
 
 
 
 m
 
 
 =
 ⟨
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ⟩
 
 
 {\displaystyle {\mathfrak {m}}=\langle x_{1},\ldots ,x_{n}\rangle }
 
 is the only homogeneous prime ideal containing 
 
 
 
 φ
 (
 I
 )
 =
 ⟨
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 k
 
 
 )
 ⟩
 .
 
 
 {\displaystyle \varphi (I)=\langle \varphi (f_{1}),\ldots ,\varphi (f_{k})\rangle .}
 
 <a href="/facts/Hilbert%27s_Nullstellensatz/TlgoRHuM">Hilbert's Nullstellensatz</a> asserts that this is the case if and only if 
 
 
 
 φ
 (
 I
 )
 
 
 {\displaystyle \varphi (I)}
 
 contains a power of each 
 
 
 
 
 x
 
 i
 
 
 ,
 
 
 {\displaystyle x_{i},}
 
 or, equivalently, that 
 
 
 
 
 
 
 m
 
 
 
 d
 
 
 ⊆
 φ
 (
 I
 )
 
 
 {\displaystyle {\mathfrak {m}}^{d}\subseteq \varphi (I)}
 
 for some positive integer d.
For this study, <a href="/facts/Francis_Sowerby_Macaulay/4XTyDBNa">Macaulay</a> introduced a matrix that is now called Macaulay matrix in degree d. Its rows are indexed by the <a href="/facts/Monomial/6VAPCYXJ">monomials</a> of degree d in 
 
 
 
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ,
 
 
 {\displaystyle x_{1},\ldots ,x_{n},}
 
 and its columns are the vectors of the coefficients on the <a href="/facts/Monomial_basis/sV6znTYv">monomial basis</a> of the polynomials of the form 
 
 
 
 m
 φ
 (
 
 f
 
 i
 
 
 )
 ,
 
 
 {\displaystyle m\varphi (f_{i}),}
 
 where m is a monomial of degree 
 
 
 
 d
 −
 deg
 ⁡
 (
 
 f
 
 i
 
 
 )
 .
 
 
 {\displaystyle d-\deg(f_{i}).}
 
 One has 
 
 
 
 
 
 
 m
 
 
 
 d
 
 
 ⊆
 φ
 (
 I
 )
 
 
 {\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, 
 
 
 
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 k
 
 
 )
 
 
 {\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}
 
 have always a non-trivial common zero.
Otherwise, let 
 
 
 
 
 d
 
 i
 
 
 
 
 {\displaystyle d_{i}}
 
 be the degree of 
 
 
 
 
 f
 
 i
 
 
 ,
 
 
 {\displaystyle f_{i},}
 
 and suppose that the indices are chosen in order that 
 
 
 
 
 d
 
 2
 
 
 ≥
 
 d
 
 3
 
 
 ≥
 ⋯
 ≥
 
 d
 
 k
 
 
 ≥
 
 d
 
 1
 
 
 .
 
 
 {\displaystyle d_{2}\geq d_{3}\geq \cdots \geq d_{k}\geq d_{1}.}
 
 The degree

D
        =
        
          d
          
            1
          
        
        +
        
          d
          
            2
          
        
        +
        ⋯
        +
        
          d
          
            n
          
        
        −
        n
        +
        1
        =
        1
        +
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        (
        
          d
          
            i
          
        
        −
        1
        )
      
    
    {\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 
 
 
 
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 k
 
 
 )
 
 
 {\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 
 
 
 
 
 
 r
 
 
 ,
 
 
 {\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 
 
 
 
 
 
 r
 
 
 
 
 {\displaystyle {\mathfrak {r}}}
 
 is a <a href="/facts/Principal_ideal/Cga66qgM">principal ideal</a> (although Macaulay matrix in degree D is not a square matrix when k > 2), which is generated by the <a href="/facts/Macaulay%27s_resultant/A5g2MdhT">resultant</a> of 
 
 
 
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 n
 
 
 )
 .
 
 
 {\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{n}).}
 
 This ideal is also <a href="/facts/Generic_property/Uz0FcCBW">generically</a> a <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a>, as it is prime if R is the ring of <a href="/facts/Integer_polynomial/Lzak8VVx">integer polynomials</a> with the all coefficients of 
 
 
 
 φ
 (
 
 f
 
 1
 
 
 )
 ,
 …
 ,
 φ
 (
 
 f
 
 k
 
 
 )
 
 
 {\displaystyle \varphi (f_{1}),\ldots ,\varphi (f_{k})}
 
 as indeterminates.

<h2 id="geometrical-interpretation">Geometrical interpretation</h2>
In the preceding formulation, the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> 
 
 
 
 R
 [
 
 x
 
 ]
 =
 R
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 
 
 {\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]}
 
 defines a morphism of <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a> (which are algebraic varieties if R is finitely generated over a field)

P
          
          
            R
          
          
            n
            −
            1
          
        
        =
        Proj
        ⁡
        (
        R
        [
        
          x
        
        ]
        )
        →
        Spec
        ⁡
        (
        R
        )
        .
      
    
    {\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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Elimination_of_quantifiers/ftDaN3Y7">Elimination of quantifiers</a></li>
<li><a href="/facts/Macaulay%27s_resultant/A5g2MdhT">Macaulay's resultant</a></li>
<li><a href="/facts/Gr%C3%B6bner_basis/efMrjM7w">Gröbner basis</a></li></ul>

<ul><li><a href="/facts/David_Mumford/WJoJ8K2u">Mumford, David</a> (1999). The Red Book of Varieties and Schemes. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783540632931.</li>
<li><a href="/facts/David_Eisenbud/hw2yEnnx">Eisenbud, David</a> (2013). Commutative Algebra: with a View Toward Algebraic Geometry. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781461253501.</li>
<li><a href="/facts/James_Milne_(mathematician)/8xYEQbw0">Milne, James S.</a> (2014). "The Work of John Tate". The Abel Prize 2008–2012. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783642394492.</li></ul>

Main theorem of elimination theory open-in-new

Main theorem of elimination theory