Standard Bases 1 Localization and Multiplicities investigating local properties 2 Singular Points on Curves and Surfaces visualization of algebraic surfaces 3 Mora’s Normal Form Algorithm a weak normal form 4 Computing Multiplicities multiplicity as the dimension of the local quotient ring MCS 563 Lecture 23 Analytic Symbolic Computation Jan Verschelde, 7 March 2013 Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 1/24 Standard Bases 1 Localization and Multiplicities investigating local properties 2 Singular Points on Curves and Surfaces visualization of algebraic surfaces 3 Mora’s Normal Form Algorithm a weak normal form 4 Computing Multiplicities multiplicity as the dimension of the local quotient ring Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 2/24 an example Consider I = hy(x − 1), z(x − 1)i⊂ Q[x, y, z]. y(x − 1)= 0 The real picture of V (I) or is below: z(x − 1)= 0 ¨ ¨¨ ¨ t t - x 0 1 ¨ ¨¨ ¨ The local dimension of V (I) at (0, 0, 0) is 1 and is 2 at (1, 0, 0). Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 3/24 localization The localization of C[x]= C[x1, x2,..., xn] at hxi = hx1, x2,..., xni is f (x) C[x] = | f , g ∈ C[x], g(0) 6= 0 . hxi g(x) Instead of at hxi we can localize from any point z ∈ Cn, not just the origin 0, shifting the coordinate system xi = yi − zi , i = 1, 2,..., n. Using local monomial orderings (Singular: ds, Ds, ls) we compute in C[x]hxi without denominators. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 4/24 local monomial orders 2 3 2 Consider the ideal I = hx1 + x1 , x2 i. 2 Factoring the first polynomial as x1 (1 + x1), we see that V (I) consists of two distinct roots. To focus on the singular solution at the origin, we look at the lowest powers of the monomials instead of the highest ones. Therefore we order the terms in the opposite order, equivalent of taking negative weights. One global term order is lexicographic, denoted by >lex. a b The corresponding local term order neglex is x >neglex x if the leftmost nonzero entry in a − b is negative. 3 2 For example: x1x2 >neglex x1 x2. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 5/24 computing multiplicities 2 3 2 3 2 For I = hx1 + x1 , x2 i a Gröbner basis leads to the initial ideal hx1 , x2 i which shows there are six solutions. Using a local term order, a local analogue to a Gröbner basis isa standard basis. 2 2 A standard basis for I gives the monomial ideal hx1 , x2 i. If #V (I) < ∞, a Gröbner basis gives the dimension of the quotient ring C[x]/I and #V (I). Monomials not in hLT (I)i define a basis for C[x]/I. Analogously to the computation of the dimension of the quotient ring via a Gröbner basis, with a standard basis we compute the multiplicity of a point. 2 3 2 For I = hx1 + x1 , x2 i, (0, 0) ∈ V (I) has multiplicity 4. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 6/24 Standard Bases 1 Localization and Multiplicities investigating local properties 2 Singular Points on Curves and Surfaces visualization of algebraic surfaces 3 Mora’s Normal Form Algorithm a weak normal form 4 Computing Multiplicities multiplicity as the dimension of the local quotient ring Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 7/24 singularities on surfaces A point z ∈ Cn on a hypersurface f (x)= 0 is singular if f (z)= 0 and ∂f also all its partial derivatives vanish: (z)= 0, i = 1, 2,..., n. ∂xi The singularity is isolated if there exists a ball around the singular point that contains no other singular point. The affine equation for the four-nodal Cayley cubic is 1 4 x 3 + 3x 2 − 3xy 2 + 3y 2 + 2 + 3 x 2 + y 2 (z − 6) − z 3 + 4z + 7z2 = 0. This surface contains exactly three lines of multiplicity one and six lines of multiplicity four. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 8/24 visualizations From the thesis of Oliver Labs (www.OliverLabs.net): On the left is the four-nodal Cayley cubic and its 9 lines. The other picture is a 16-nodal Kummer surface, known to Kummer in 1864, a quartic surface in 3-space with the maximum number of nodes. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 9/24 a cuspidal cubic 3 2 1 1 f = x − y = 0 ∩ x + 2 y = 0 and ∩ x + 2 y = ǫ: 1.0 0.5 K1.0 K0.5 0 0.5 1.0 K0.5 K1.0 Restricting f to the shifted line 1 1 1 3 f (x = ǫ − y, y) = (ǫ − y)3 − y 2 = −y 2( y + 1)+ y 2ǫ + O(ǫ2) 2 2 8 4 and by the two zeroes close to (0, 0): multiplicity 2. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 10/24 Standard Bases 1 Localization and Multiplicities investigating local properties 2 Singular Points on Curves and Surfaces visualization of algebraic surfaces 3 Mora’s Normal Form Algorithm a weak normal form 4 Computing Multiplicities multiplicity as the dimension of the local quotient ring Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 11/24 a weak normal form ∞ Instead of x = x k (x − x 2) = ! Xk 0 we write (1 − x)x = x − x 2. This gives rise to a weak normal form: s uf = ai gi + NF(f |G), G = {g1, g2,... gs}. = Xi 1 We define ecart(f ) := deg(f ) − deg(LM(f )). For a homogeneous polynomial f we have ecart(f )= 0. Mora’s normal form algorithm returns a weak normal form of a polynomial with respect to a set of polynomials. This algorithm is analoguous to the division algorithm. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 12/24 Mora’s NF algorithm NFMora(f , G,>) Input: f ∈ C[x], G = {g1, g2,..., gs}, gi ∈ C[x], i = 1, 2,..., s; > is any monomial ordering. s Output: h ∈ C[x] : uf = ai gi + h. i=1 h := f ; T := G; X while (h 6= 0) do Th := { g ∈ T : LM(g)|LM(h) }; if Th = ∅ then return h; else choose g ∈ Th with minimal ecart(g); if ecart(g) > ecart(h) then T := T ∪ {h}; h := Spolynomial(h, g); end if; end while. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 13/24 local membership Proposition Consider I ⊂ C[x] and 0 ∈ Cn is on an irreducible component W of V (I). LetG = {g1, g2,..., gs} be a standard basis of I with respect to some local order >. Consider p ∈ C[x] and let r = NFMora(p, G,>). If r = 0, then p ∈ I(W ). Proof. If r = 0, then up = a1g1 + a2g2 + · · · + asgs, where u ∈ C[x] is invertible in C[x]hxi, i.e.: u(0) 6= 0. We have: W ⊂ V (I) ⇒ up ∈ I(W ). Because W is irreducible (I(W ) is prime): up ∈ I(W ) ⇒ u ∈ I(W ) or p ∈ I(W ). But since u(0) 6= 0, u 6= I(W ), thus p ∈ I(W ). Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 14/24 algorithm for a standard basis Standard(G, NF,>) Input: G is a finite list of polynomials, NF is an algorithm to compute a weak normal form > is any monomial ordering. Output: S is a standard basis for hGi. S := G; P := { (f , g) | f , g ∈ S, f 6= g }; while P 6= ∅ do (f , g) := pop from P; h := NF(Spolynomial(f , g), S,>); if h 6= 0 then P := P ∪ { (h, f ) | f ∈ S }; S := S ∪ {h}; end if; end while. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 15/24 the tangent cone Consider a point p ∈ Cn and f ∈ C[x], d = deg(f ). f as Taylor series about p, |a| = a1 + a2 + · · · + an, a!= a1!a2! · · · an!: 1 af n ∂ ai f (x) = f (p)+ (p) (xi − pi ) a! a1 a2 an ∂x1 ∂x2 · · · ∂xn = 0<X|a|≤d Yi 1 = fp,0 + fp,1 + · · · + fp,d . Define fp,min as fp,min(p) 6= 0 and for all j < min: fp,j (p)= 0. The tangent cone of a set S at p is the variety Cp(S)= V (fp,min | f ∈ I(S)). Standard bases compute tangent cones. Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 16/24 Standard Bases 1 Localization and Multiplicities investigating local properties 2 Singular Points on Curves and Surfaces visualization of algebraic surfaces 3 Mora’s Normal Form Algorithm a weak normal form 4 Computing Multiplicities multiplicity as the dimension of the local quotient ring Analytic Symbolic Computation (MCS 563) Standard Bases L-237March2014 17/24 intersection multiplicity Let z ∈ Cn be an isolated solution of f (x)= 0. The intersection multiplicity µ(z) is defined algebraically as the dimension of the local quotient ring: C C µ(z)= dim [x]hx1−z1,x2−z2,...,xn−zni/hf i . In analogy with global quotient rings, consider hx 2, xy, y2i: r6 r r r t r r r d t r r d d t r- The big black dots are the generators x 2, xy, and y 2. The small black dots are the generated monomials. The monomials 1, x, and y represented by the empty circles are a basis for the quotient ring. The multiplicity of (0, 0) equals three, the number of monomials under the staircase.