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Discriminant of Symmetric Homogeneous Polynomials

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Discriminant of Symmetric Homogeneous Polynomials Discriminant of symmetric homogeneous polynomials Laurent Bus´e INRIA Sophia Antipolis, France [email protected] Seminar of Algebraic Geometry, University of Barcelona October 3, 2014 Joint work with Anna Karasoulou (University of Athens) Outline 1 Tools from elimination theory • Discriminant of a hypersurface in a projective space • Resultant of homogeneous polynomials 2 The main formula • Homogeneous symmetric polynomials • Divided differences • Decomposition of the discriminant of a homogeneous symmetric polynomial 3 Sn-equivariant systems • Sn-equivariant systems of homogeneous polynomials • Resultant of a Sn-equivariant polynomial system The universal discriminant Disc belongs to An;d n−1 I It is homogeneous of degree n(d − 1) ( usual grading of An;d ). I It is a prime element in An;d (irreducible in An;d ⊗ Q and not divisible by any integer > 1). Discriminant of a hypersurface in a projective space Notation: I Fix integers n ≥ 1, d ≥ 2 and let k be a commutative ring. I Pd (k) := Γ(O n−1 (d)) = k[x1;:::; xn]d (Rk : Pd (k) = Pd ( ) ⊗ k). Pk Z Z I The universal homogeneous polynomial of degree d in n variables: X α α α1 α2 αn Pn;d (x1;:::; xn) = Uαx (x = x1 x2 ··· xn ): jαj=d I The universal coefficient ring : An;d := Z[Uα : jαj = d]. Discriminant of a hypersurface in a projective space Notation: I Fix integers n ≥ 1, d ≥ 2 and let k be a commutative ring. I Pd (k) := Γ(O n−1 (d)) = k[x1;:::; xn]d (Rk : Pd (k) = Pd ( ) ⊗ k). Pk Z Z I The universal homogeneous polynomial of degree d in n variables: X α α α1 α2 αn Pn;d (x1;:::; xn) = Uαx (x = x1 x2 ··· xn ): jαj=d I The universal coefficient ring : An;d := Z[Uα : jαj = d]. The universal discriminant Disc belongs to An;d n−1 I It is homogeneous of degree n(d − 1) ( usual grading of An;d ). I It is a prime element in An;d (irreducible in An;d ⊗ Q and not divisible by any integer > 1). Smoothness criterion For any algebraically closed field k and polynomial f 2 Pd (k) @f @f @f Disc(f ) 6= 0 , V n−1 f ; ; ;:::; = ;: Pk @x1 @x2 @xn Resultant of the partial derivatives of F Discriminant of a hypersurface in a projective space Specialization : For any ring k and polynomial f 2 Pd (k) An;d ! k (canonical ring map) u(:::; Uα;:::) 7! u(f ) Remark : Pn;d 2 Pd (An;d ) and u(Pn;d ) = u Definition (discriminant) The discriminant of f 2 Pd (k) is Disc(f ) 2 k. Discriminant of a hypersurface in a projective space Specialization : For any ring k and polynomial f 2 Pd (k) An;d ! k (canonical ring map) u(:::; Uα;:::) 7! u(f ) Remark : Pn;d 2 Pd (An;d ) and u(Pn;d ) = u Definition (discriminant) The discriminant of f 2 Pd (k) is Disc(f ) 2 k. Smoothness criterion For any algebraically closed field k and polynomial f 2 Pd (k) @f @f @f Disc(f ) 6= 0 , V n−1 f ; ; ;:::; = ;: Pk @x1 @x2 @xn Resultant of the partial derivatives of F Resultant of homogeneous polynomials Notation: I Fix integers n ≥ 1, d1;:::; dn ≥ 1. I The universal homogeneous polynomial of degree di in n variables: X α Pn;di (x1;:::; xn) = Ui,αx (i = 1;:::; n): jαj=di I The universal coefficient ring : Ad1;:::;dn := Z[Ui,α : i = 1;:::; n jαj = di ]: The universal resultant Res belongs to Ad1;:::;dn I It is homogeneous w.r.t. Pn;di of degree d1 ::: dn=di . I It is a prime element in Ad1;:::;dn . Resultant can be explicitly computed I closed matrix-based formulas (Sylvester, Cayley, Macaulay, etc) I many formal properties (multiplicativity, change of basis, etc) Resultant of homogeneous polynomials Specialization : For any ring k and polynomials fi 2 Pdi (k), i = 1;:::; n Ad1;:::;dn ! k (canonical ring map) u(:::; Ui,α;:::) 7! u(f1;:::; fn) Definition (resultant) The resultant of f1;:::; fn is Res(f1;:::; fn) 2 k. d1 d2 dn Remark : resultant is normalized by Res(x1 ; x2 ;:::; xn ) = 1. Common root detection For any algebraically closed field k and polynomials fi 2 Pdi (k) Res(f ;:::; f ) 6= 0 , V n (f ;:::; f ) = ;: 1 n Pk 1 n Resultant of homogeneous polynomials Specialization : For any ring k and polynomials fi 2 Pdi (k), i = 1;:::; n Ad1;:::;dn ! k (canonical ring map) u(:::; Ui,α;:::) 7! u(f1;:::; fn) Definition (resultant) The resultant of f1;:::; fn is Res(f1;:::; fn) 2 k. d1 d2 dn Remark : resultant is normalized by Res(x1 ; x2 ;:::; xn ) = 1. Common root detection For any algebraically closed field k and polynomials fi 2 Pdi (k) Res(f ;:::; f ) 6= 0 , V n (f ;:::; f ) = ;: 1 n Pk 1 n Resultant can be explicitly computed I closed matrix-based formulas (Sylvester, Cayley, Macaulay, etc) I many formal properties (multiplicativity, change of basis, etc) Example of the Clebsch surface : P4 3 P4 3 I f (x1;:::; x4) = i=1 xi − ( i=1 xi ) 37 I Res(@1f ;:::;@4f ) = −3 × 5 I a(4; 3) = 15=3 = 5 32 I Disc(f ) = −3 × 5 3 I But 3 divides f and Disc is homogeneous of degree 4 × 2 = 32 I Therefore Disc(f =3) = −5 smooth expect in characteristic 5. Computation of the discriminant Theorem/Definition (Demazure) For any polynomial f 2 Pd (k) @f @f @f d a(n;d)Disc(f ) = Res ; ;:::; @x1 @x2 @xn where (d − 1)n − (−1)n a(n; d) := 2 : d Z Computation of the discriminant Theorem/Definition (Demazure) For any polynomial f 2 Pd (k) @f @f @f d a(n;d)Disc(f ) = Res ; ;:::; @x1 @x2 @xn where (d − 1)n − (−1)n a(n; d) := 2 : d Z Example of the Clebsch surface : P4 3 P4 3 I f (x1;:::; x4) = i=1 xi − ( i=1 xi ) 37 I Res(@1f ;:::;@4f ) = −3 × 5 I a(4; 3) = 15=3 = 5 32 I Disc(f ) = −3 × 5 3 I But 3 divides f and Disc is homogeneous of degree 4 × 2 = 32 I Therefore Disc(f =3) = −5 smooth expect in characteristic 5. 2 I Example n = d = 2. P2;2 = U2x2 + U1xy + U0y , 2 Disc(P2;2) = 4U2U0 − U1 ; Disc(U0 = U2) = (2U0 − U1)(2U0 + U1): 3 2 2 3 I Example n = 2, d = 3. P2;3 = U3x + U2x y + U1xy + U0y , 2 2 3 2 2 3 Disc(P2;2) = 27 U3 U0 +4 U3U1 −18 U3U2U0U1−U1 U2 +4 U2 U0; 3 Disc(U3 = U0; U2 = U1) = (U0 + U1)(3 U0 − U1) : Main observation I The discriminant of the generic homogeneous polynomial is irreducible in the universal ring of coefficients. I However, it always factorizes with symmetric polynomials, I and this leads to an easier computation (useful in various applications). Main observation I The discriminant of the generic homogeneous polynomial is irreducible in the universal ring of coefficients. I However, it always factorizes with symmetric polynomials, I and this leads to an easier computation (useful in various applications). 2 I Example n = d = 2. P2;2 = U2x2 + U1xy + U0y , 2 Disc(P2;2) = 4U2U0 − U1 ; Disc(U0 = U2) = (2U0 − U1)(2U0 + U1): 3 2 2 3 I Example n = 2, d = 3. P2;3 = U3x + U2x y + U1xy + U0y , 2 2 3 2 2 3 Disc(P2;2) = 27 U3 U0 +4 U3U1 −18 U3U2U0U1−U1 U2 +4 U2 U0; 3 Disc(U3 = U0; U2 = U1) = (U0 + U1)(3 U0 − U1) : Example : formula in the case n ≥ d = 2 The generic homogeneous symmetric polynomial is of the form 0 1 n !2 Y X F = c(2) @ xi xj A + c(1;1) xi 1≤i<j≤n i=1 and its discriminant is given by ( n−1 −c(2) (n − 1)c(2) + 2nc(1;1) if n is even; Disc(F ) = n−1 n−1 c(2) 2 c(2) + nc(1;1) if n is odd : Goal of this talk Our goal is to give a universal closed formula for the discriminant of a homogeneous symmetric polynomial I First attempt by Perminov and Shakirov (arXiv:0910.5757) I The main difficulty comes from the multiplicity of the factors Goal of this talk Our goal is to give a universal closed formula for the discriminant of a homogeneous symmetric polynomial I First attempt by Perminov and Shakirov (arXiv:0910.5757) I The main difficulty comes from the multiplicity of the factors Example : formula in the case n ≥ d = 2 The generic homogeneous symmetric polynomial is of the form 0 1 n !2 Y X F = c(2) @ xi xj A + c(1;1) xi 1≤i<j≤n i=1 and its discriminant is given by ( n−1 −c(2) (n − 1)c(2) + 2nc(1;1) if n is even; Disc(F ) = n−1 n−1 c(2) 2 c(2) + nc(1;1) if n is odd : The generic homogeneous symmetric polynomial of degree d X F (x1;:::; xn) := cλeλ(x) 2 Z[cλ : λ ` d][x1;:::; xn]: λ`d I Example for d = 2 2 F (x1;:::; xn) = c(2)e2(x1;:::; xn) + c(1;1)e1 (x1;:::; xn): Homogeneous symmetric polynomials I Elementary symmetric polynomials in x = x1;:::; xn n X p Y ep(x)t = (1 + xi t) p≥0 i=1 I Given a partition λ = (λ1 ≥ · · · ≥ λk ), define eλ(x) := eλ1 (x)eλ2 (x) ··· eλk (x) 2 Z[x1;:::; xn]: Pn I If i=1 λi = d we say that λ is a partition of d , denoted λ ` d.
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