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Elimination Theory Elimination Theory Carlos D'Andrea Fourth EACA International School on Computer Algebra and its Applications Carlos D'Andrea Elimination Theory What is Elimination Theory? Carlos D'Andrea Elimination Theory What is Elimination Theory? Carlos D'Andrea Elimination Theory Elimination Theory Carlos D'Andrea Elimination Theory Elimination Theory Carlos D'Andrea Elimination Theory Elimination Theory Carlos D'Andrea Elimination Theory Elimination Theory Carlos D'Andrea Elimination Theory In the problem of elimination, one seeks the relationship that must exist between the coefficients of a function or system of functions in order that some particular circumstance (or singularity) can occur. Cayley, 1864 Nouvelles recherches sur l'´elimination et la th´eorie des courbes A bit of history Carlos D'Andrea Elimination Theory A bit of history In the problem of elimination, one seeks the relationship that must exist between the coefficients of a function or system of functions in order that some particular circumstance (or singularity) can occur. Cayley, 1864 Nouvelles recherches sur l'´elimination et la th´eorie des courbes Carlos D'Andrea Elimination Theory A bit of history A system of arbitrarily many algebraic equations for z0; z0; z00; :::; z(n−1), in which the coefficients belong to the rationality domain( R; R0; R00;::::), define the algebraic relations between z and R, whose knowledge and representation are the purpose of the theory of elimination. Kronecker, 1882 Grundz¨ugeeiner arithmetischen Theorie der algebraischen Gr¨ossen Carlos D'Andrea Elimination Theory Find \the condition" on a10; a11; a20; a21 so that the system a10x0 + a11x1 = 0 a20x0 + a21x1 = 0 has a solution different from (0; 0) The baby example in elimination Carlos D'Andrea Elimination Theory The baby example in elimination Find \the condition" on a10; a11; a20; a21 so that the system a10x0 + a11x1 = 0 a20x0 + a21x1 = 0 has a solution different from (0; 0) Carlos D'Andrea Elimination Theory # a10a21 − a20a11 2 K[a10; a11; a20; a21] \Elimination" a10x0 + a11x1; a20x0 + a21x1 2 K[a10; a11; a20; a21; x0; x1] Carlos D'Andrea Elimination Theory \Elimination" a10x0 + a11x1; a20x0 + a21x1 2 K[a10; a11; a20; a21; x0; x1] # a10a21 − a20a11 2 K[a10; a11; a20; a21] Carlos D'Andrea Elimination Theory to have a solution different from (0; 0;:::; 0) More general Find \the condition" for the system 8 > a10x0 + a11x1 + ::: + a1nxn = 0 > < a20x0 + a21x1 + ::: + a2nxn = 0 . > . > : a(n+1)0x0 + a(n+1)1x1 + ::: + a(n+1)nxn = 0 Carlos D'Andrea Elimination Theory More general Find \the condition" for the system 8 > a10x0 + a11x1 + ::: + a1nxn = 0 > < a20x0 + a21x1 + ::: + a2nxn = 0 . > . > : a(n+1)0x0 + a(n+1)1x1 + ::: + a(n+1)nxn = 0 to have a solution different from (0; 0;:::; 0) Carlos D'Andrea Elimination Theory Another more general Let d1; d2 2 N: Find \the condition" for the system of polynomials d d −1 a10x0 1 + a11x0 1 x1 + ::: = 0 d d −1 a20x0 2 + a21x0 2 x1 + ::: = 0 to have a solution different from (0; 0) Carlos D'Andrea Elimination Theory More more more general... Let n 2 N; and d1;:::; dn+1 2 N; find the condition for 8 P α0 αn a1,α ,...,α x0 ::: xn = 0 > α0+:::+αn=d1 0 n > > > P α0 αn > a2,α ,...,α x0 ::: xn = 0 <> α0+:::+αn=d2 0 n . > . > > > :> P a x α0 ::: x αn = 0 α0+:::+αn=dn+1 n+1,α0,...,αn 0 n to have a solution different from (0; 0;:::; 0) Carlos D'Andrea Elimination Theory Elimination: The general problem For a = (a1;:::; aN ); k; n 2 N let f1(a; x1;:::; xn);:::; fk (a; x1;:::; xn) 2 K[a; x1;:::; xn]: Find conditions on a such that 8 > f1(a; x1;:::; xn) = 0 > < f2(a; x1;:::; xn) = 0 . > . > : fk (a; x1;:::; xn) = 0 has a solution Carlos D'Andrea Elimination Theory There is not necessarily a \closed" condition Algorithms?? Solution? Depends on the ground field Carlos D'Andrea Elimination Theory Algorithms?? Solution? Depends on the ground field There is not necessarily a \closed" condition Carlos D'Andrea Elimination Theory Solution? Depends on the ground field There is not necessarily a \closed" condition Algorithms?? Carlos D'Andrea Elimination Theory Conditions: all maximal minors of aij 1≤i≤k; 1≤j≤n equal to zero Easy example 8 > a11x1 + ::: + a1nxn = 0 > < a21x1 + ::: + a2nxn = 0 . > . > : aknx1 + ::: + aknxn = 0 with k ≥ n Carlos D'Andrea Elimination Theory Easy example 8 > a11x1 + ::: + a1nxn = 0 > < a21x1 + ::: + a2nxn = 0 . > . > : aknx1 + ::: + aknxn = 0 with k ≥ n Conditions: all maximal minors of aij 1≤i≤k; 1≤j≤n equal to zero Carlos D'Andrea Elimination Theory Conditions? Another \easy" example k = n = 1; 2 d a0 + a1x1 + a2x1 + ::: + ad x1 = 0 Carlos D'Andrea Elimination Theory Another \easy" example k = n = 1; 2 d a0 + a1x1 + a2x1 + ::: + ad x1 = 0 Conditions? Carlos D'Andrea Elimination Theory V ⊂ KN × Kn π1jV ## π1 N π1(V ) ⊂ K The set of conditions is π1(V ) is not necessarily described by zeroes of polynomials Geometry V = f(a; x1;:::; xn): f1(a; x1;:::; xn) = 0;::: fk (a; x1;:::; xn) = 0g Carlos D'Andrea Elimination Theory The set of conditions is π1(V ) is not necessarily described by zeroes of polynomials Geometry V = f(a; x1;:::; xn): f1(a; x1;:::; xn) = 0;::: fk (a; x1;:::; xn) = 0g V ⊂ KN × Kn π1jV ## π1 N π1(V ) ⊂ K Carlos D'Andrea Elimination Theory Geometry V = f(a; x1;:::; xn): f1(a; x1;:::; xn) = 0;::: fk (a; x1;:::; xn) = 0g V ⊂ KN × Kn π1jV ## π1 N π1(V ) ⊂ K The set of conditions is π1(V ) is not necessarily described by zeroes of polynomials Carlos D'Andrea Elimination Theory V = f(a; x0; x1;:::; xn): f1(a; x0; x1;:::; xn) = 0;::: fk (a; x0; x1;:::; xn) = 0g V ⊂ KN × Pn π1jV ## π1 N π1(V ) ⊂ K π1(V ) = fp1(a) = 0;:::; p`(a) = 0g Projective Elimination (homogeneous polynomials in an algebraically closed field) Carlos D'Andrea Elimination Theory V ⊂ KN × Pn π1jV ## π1 N π1(V ) ⊂ K π1(V ) = fp1(a) = 0;:::; p`(a) = 0g Projective Elimination (homogeneous polynomials in an algebraically closed field) V = f(a; x0; x1;:::; xn): f1(a; x0; x1;:::; xn) = 0;::: fk (a; x0; x1;:::; xn) = 0g Carlos D'Andrea Elimination Theory π1(V ) = fp1(a) = 0;:::; p`(a) = 0g Projective Elimination (homogeneous polynomials in an algebraically closed field) V = f(a; x0; x1;:::; xn): f1(a; x0; x1;:::; xn) = 0;::: fk (a; x0; x1;:::; xn) = 0g V ⊂ KN × Pn π1jV ## π1 N π1(V ) ⊂ K Carlos D'Andrea Elimination Theory Projective Elimination (homogeneous polynomials in an algebraically closed field) V = f(a; x0; x1;:::; xn): f1(a; x0; x1;:::; xn) = 0;::: fk (a; x0; x1;:::; xn) = 0g V ⊂ KN × Pn π1jV ## π1 N π1(V ) ⊂ K π1(V ) = fp1(a) = 0;:::; p`(a) = 0g Carlos D'Andrea Elimination Theory Input: f1(a; x);::: fk(a; x) Output: p1(a);:::; p`(a) How do we compute these conditions? Carlos D'Andrea Elimination Theory Output: p1(a);:::; p`(a) How do we compute these conditions? Input: f1(a; x);::: fk(a; x) Carlos D'Andrea Elimination Theory How do we compute these conditions? Input: f1(a; x);::: fk(a; x) Output: p1(a);:::; p`(a) Carlos D'Andrea Elimination Theory Set s := max deg(fi ), and G0;:::; Gn expressions of the form P α Gj (a; x) := uα;i x fi (a; x) of degree s in x with new parameters uα;i . Then, \the Kronecker u-Resultant" P β Resx (G0;:::; Gn) = β Pβ(a)u gives the conditions fPβ(a)g Mertens 1889 Carlos D'Andrea Elimination Theory Then, \the Kronecker u-Resultant" P β Resx (G0;:::; Gn) = β Pβ(a)u gives the conditions fPβ(a)g Mertens 1889 Set s := max deg(fi ), and G0;:::; Gn expressions of the form P α Gj (a; x) := uα;i x fi (a; x) of degree s in x with new parameters uα;i . Carlos D'Andrea Elimination Theory Mertens 1889 Set s := max deg(fi ), and G0;:::; Gn expressions of the form P α Gj (a; x) := uα;i x fi (a; x) of degree s in x with new parameters uα;i . Then, \the Kronecker u-Resultant" P β Resx (G0;:::; Gn) = β Pβ(a)u gives the conditions fPβ(a)g Carlos D'Andrea Elimination Theory For arbitrary s, we use the \Macaulay matrix" Ms(a) such that 2 . 3 2 . 3 . β α Ms(a) 6 x 7 = 6 x fi 7 ; jαj = s 4 . 5 4 . 5 . For s 0, all maximal minors of Ms(a) vanish iff there is a common solution van der Waerden 1926 Carlos D'Andrea Elimination Theory For s 0, all maximal minors of Ms(a) vanish iff there is a common solution van der Waerden 1926 For arbitrary s, we use the \Macaulay matrix" Ms(a) such that 2 . 3 2 . 3 . β α Ms(a) 6 x 7 = 6 x fi 7 ; jαj = s 4 . 5 4 . 5 . Carlos D'Andrea Elimination Theory van der Waerden 1926 For arbitrary s, we use the \Macaulay matrix" Ms(a) such that 2 . 3 2 . 3 . β α Ms(a) 6 x 7 = 6 x fi 7 ; jαj = s 4 . 5 4 . 5 . For s 0, all maximal minors of Ms(a) vanish iff there is a common solution Carlos D'Andrea Elimination Theory I π1(V ) = 1 (hf1;:::; fki : hx0;:::; xni ) \ K[a] The \elimination ideal" Algebra & Geometry (van der Waerden, 1926) Carlos D'Andrea Elimination Theory \ K[a] The \elimination ideal" Algebra & Geometry (van der Waerden, 1926) I π1(V ) = 1 (hf1;:::; fki : hx0;:::; xni ) Carlos D'Andrea Elimination Theory Algebra & Geometry (van der Waerden, 1926) I π1(V ) = 1 (hf1;:::; fki : hx0;:::; xni ) \ K[a] The \elimination ideal" Carlos D'Andrea Elimination Theory the polynomials are not homogeneous? K is not algebraically closed? But what if..
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