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Primary Decomposition p. Primary Decomposition Gerhard Pfister pfister@mathematik.uni-kl.de Departement of Mathematics University of Kaiserslautern

Primary Decomposition p. Primary Decomposition:References Gianni, P.; Trager, B.; Zacharias, G.: Gröbner Bases and Primary Decomposition of Polynomial Ideals. J. Symb. Comp. 6, 149 167 (1988). Eisenbud, D.; Huneke, C.; Vasconcelos, W.: Direct Methods for Primary Decomposition. Invent. Math. 110, 207 235 (1992). Shimoyama, T.; Yokoyama, K.: Localization and Primary Decomposition of Polynomial ideals. J. Symb. Comp. 22, 247 277 (1996). Decker, W.; Greuel, G.-M.; Pfister, G.: Primary Decomposition: Algorithms and Comparisons. In: Algorithmic Algebra and Number Theory, Springer, 187 220 (1998).

Primary Decomposition p. Gianni, Trager, Zacharias Definition A maximal ideal M K[x 1,..., x n ] is called in general position with respect to the lexicographical ordering with x 1 > > x n, if there exist g 1,..., g n K[x n ] with M = x 1 + g 1 (x n ),..., x n 1 + g n 1 (x n ), g n (x n ). A zero dimensional ideal I K[x 1,..., x n ] is called in general position with respect to the lexicographical ordering with x 1 > > x n, if all associated primes P 1,..., P k are in general position and if P i K[x n ] P j K[x n ] for i j.

Primary Decomposition p. Proposition Let K be a field of characteristic 0, and let I K[x], x = (x 1,..., x n ), be a zero dimensional ideal. Then there exists a non empty, Zariski open subset U K n 1 such that for all a = (a 1,..., a n 1 ) U, the coordinate change ϕ a : K[x] K[x] defined by ϕ a (x i ) = x i if i < n, and ϕ a (x n ) = x n + n 1 i=1 a i x i has the property that ϕ a (I) is in general position with respect to the lexicographical ordering defined by x 1 > > x n.

Primary Decomposition p. Proposition Let I K[x 1,..., x n ] be a zero dimensional ideal. Let g = I K[x n ], g = g ν 1 1... gν s s, g i monic and prime and g i g j for i j. Then I = s i=1 I, gν i i.

Primary Decomposition p. Proposition Let I K[x 1,..., x n ] be a zero dimensional ideal. Let g = I K[x n ], g = g ν 1 1... gν s s, g i monic and prime and g i g j for i j. Then I = s i=1 I, gν i i. If I is in general position with respect to the lexicographical ordering with x 1 > > x n, then (2) I, g ν i i is a primary ideal for all i.

Primary Decomposition p. Criterion Let I K[x 1,..., x n ] be a proper ideal. Then the following conditions are equivalent: I is zero dimensional, primary and in general position with respect to the lexicographical ordering with x 1 > > x n. There exist g 1,..., g n K[x n ] and positive integers ν 1,..., ν n such that I K[x n ] = g ν n n, g n irreducible; for each j < n, I contains the element `x j + g j νj. Let S be a reduced Gröbner basis of I with respect to the lexicographical ordering with x 1 >... > x n. Then there exist g 1,..., g n K[x n ] and positive integers ν 1,..., ν n such that g ν n n S and g n is irreducible; (x j + g j ) ν j is congruent to an element in S K[x j,..., x n ] modulo g n, x n 1 + g n 1,..., x j+1 + g j+1 K[x] for j = 1,..., n 1.

Primary Decomposition p. primarytest(i) Input: A zero dimensional ideal I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: I if I is primary and in general position or < 0 > else. compute a reduced Gröbner basis S of I with respect to the lexicographical ordering with x 1 > > x n ; factorize g S, the element with smallest leading monomial; if (g = g ν n n with g n irreducible) prim := g n else return 0. i := n; while (i > 1) i := i 1; choose f S with LM(f) = x m i ; b := the coefficient of x m 1 i polynomial in x i ; q := x i + b/m; if (q m f mod prim) else return 0 ; return prim. in f considered as prim := prim + q ;

return result. Primary Decomposition p. zerodecomp(i) Input: a zero-dimensional ideal I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: a set of pairs (Q i, P i ) of ideals in K[x], i = 1,..., r, such that I = Q 1 Q r is a primary decomposition of I, and P i = Q i, i = 1,..., r. result := ; choose a random a K n 1, and apply the coordinate change I := ϕ a (I); compute a Gröbner basis G of I with respect to the lexicographical ordering with x 1 > > x n, let g G be the element with smallest leading monomial. factorize g = g ν 1 1... gν s s K[x n ]; for i = 1 to s do set Q i := I, g ν i i and Q i := I, ϕ 1 a (g i ) ν i ; set P i := PRIMARYTEST(Q i ); if P i 0 set P i := ϕ 1 a (P i ); result := result {(Q i, P i )}; else result := result ZERODECOMP (Q i );

Primary Decomposition p. Proposition Let I K[x] be an ideal and u x = {x 1,..., x n } be a maximal independent set of variables with respect to I. (I K[u] = {0} and #(u) = dim(k[x]/i)) IK(u)[x u] K(u)[x u] is a zero dimensional ideal. Let S = {g 1,..., g s } I K[x] be a Gröbner basis of IK(u)[x u], and let h := lcm ( LC(g 1 ),..., LC(g s ) ) K[u], then IK(u)[x u] K[x] = I : h, and this ideal is equidimensional of dimension dim(i).

Primary Decomposition p. 1 reductiontozero(i) Input: I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: A list (u, G, h), where u x is a maximal independent set with respect to I, G = {g 1,..., g s } I is a Gröbner basis of IK(u)[x u], h K[u] such that IK(u)[x u] K[x] = I : h = I : h. compute a maximal independent set u x with respect to I; compute a Gröbner basis G = {g 1,..., g s } of I with respect to the lexicographical ordering with x u > u; h := Q s i=1 LC(g i) K[u], where the g i are considered as polynomials in x u with coefficients in K(u); compute m such that g 1,..., g s : h m = g 1,..., g s : h m+1 ; return u, {g 1,..., g s }, h m.

Primary Decomposition p. 1 decomp(i) Input: I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: a set of pairs (Q i, P i ) of ideals in K[x], i = 1,..., r, such that I = Q 1 Q r is a primary decomposition of I, and P i = p (Q i ), i = 1,..., r. `u, G, h := REDUCTIONTOZERO (I); change ring to K(u)[x u] and compute qprimary := ZERODECOMP ( G K(u)[x u] ); change ring to K[x] and compute primary := {(Q K[x], P K[x]) (Q, P ) qprimary}; primary := primary DECOMP ( I, h n ); return primary.

Primary Decomposition p. 1 Definition Let A be a Noetherian ring, let I A be an ideal, and let I = Q 1 Q s be an irredundant primary decomposition. The equidimensional part E(I) is the intersection of all primary ideals Q i with dim(q i ) = dim(i). The ideal I (respectively the ring A/I) is called equidimensional or pure dimensional if E(I) = I. In particular, the ring A is called equidimensional if E( 0 ) = 0.

Primary Decomposition p. 1 equidimensional(i) Input: I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: E(I) K[x], the equidimensional part of I. set (u, G, h) := REDUCTIONTOZERO (I); if (dim( I, h ) < dim(i)) return ( G : h ); else return ( ( G : h ) EQUIDIMENSIONAL ( I, h ) ).

Primary Decomposition p. 1 Proposition Let I K[x 1,..., x n ] be a zero dimensional ideal and I K[x i ] = f i for i = 1,..., n. Moreover, let g i be the squarefree part of f i, then I = I + g 1,..., g n.

Primary Decomposition p. 1 proof Obviously, I I + g 1,..., g n I. Hence, it remains to show that a n I implies that a I + g 1,..., g n.

Primary Decomposition p. 1 proof Obviously, I I + g 1,..., g n I. Hence, it remains to show that a n I implies that a I + g 1,..., g n. Let K be the algebraic closure of K. We see that each g i is the product of different linear factors of K[x i ]. These linear factors of the g i induce a splitting of the ideal (I + g 1,..., g n )K[x] into an intersection of maximal ideals.

Primary Decomposition p. 1 zeroradical(i) Input: a zero dimensional ideal I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: I K[x], the radical of I. for i = 1,..., n, compute f i K[x i ] such that I K[x i ] = f i ; return I + SQUAREFREE (f 1 ),..., SQUAREFREE (f n ).

Primary Decomposition p. 1 radical(i) Input: I := f 1,..., f k K[x], x = (x 1,..., x n ). Output: I K[x], the radical of I. (u, G, h) := REDUCTIONTOZERO (I); change ring to K(u)[x u] and compute J := ZERORADICAL ( G ); compute a Gröbner basis {g 1,..., g l } K[x] of J; set p := l i=1 LC(g i) K[u]; change ring to K[x] and compute J K[x] = g 1,..., g l : p ; return (J K[x]) RADICAL ( I, h ).

Primary Decomposition p. 1 Hensel s Lemma Let A be one of the following rings: Z, Z[x 1,..., x n ], Q[x 1,..., x n ], C[x 1,..., x n ]. Let I A be an ideal and f(x) A[x] monic. Assume, g 1 (x), h 1 (x) A/I[x] are relatively prime and monic, such that f(x) = g 1 (x) h 1 (x) mod I. Then there exist monic polynomials g n, h n A/I n [x] such that f = g n h n mod I n g n = g 1 mod I, h n = h 1 mod I

Primary Decomposition p. 1 Lifting a factorization f C[x 1,..., x n ] I = x 3 a 3,..., x n a N, d i = deg xi (f) f (i) = f(x 1,..., x i, a 4,..., a n ) f (2) = g 1 h 1 Hensel s lemma in A[x 1 ] (A = C[x 2, x 3 ], I = x 3 a 3 ) f (3) = g d3+1 h d3+1 mod x 3 a 3 d 3+1 if f = f 1 f 2 and f 1 (x 1, x 2, a 3,..., a n ) = g 1, f 2 (x 1, x 2, a 3,..., a n ) = h 1 then f 1 (x 1, x 2, x 3, a 4... a n ) = g d3+1 (x 1, x 2, x 3 ) f 2 (x 1, x 2, x 3, a 4... a n ) = h d3+1 (x 1, x 2, x 3 ) by unicity of Hensel s lemma. Restart with the next variable.

Primary Decomposition p. 2 Gao s version of Bertini s Theorem Let K be a field of characteristic 0 and S K a finite subset. Let f K[x 1,..., x n ], deg(f) = d and f 0 (x, y) = f(a 1 x + b 1 y + c 1,..., a n x + b n y + c n ) Then, for random choices of a i, b i, c i in S with probability at least 1 2d3 S all the absolute irreducible factors of f remain absolutely irreducible factors of f 0 in K[x, y].

Primary Decomposition p. 2 Irreducibility Testing Let f Z[x, y] be irreducible, if for some prime p f is irreducible in Z/pZ[x, y] there exists a simple point (a, b) (Z/pZ) 2 of V (f) the degree of f mod p is equal to the degree of f. The test is based on the following theorem: Let k be a field and (α, β) k 2 be a simple point of f k[x, y]. Then one absolute irreducible factor belongs to k[α, β][x, y].

Primary Decomposition p. 2 Splitting over C Theorem: Gao/Ruppert Let f Q[x, y] be irreducible of bidegree (m, n). Let G = {g Q[x, y] (m 1, n) deg(g), h Q[x, y], (g/f) y The vector space G has the following properties f is irreducible in C[x, y] if and only if dim Q (G) = 1. gg f x G mod f for all g G. Let g 1,..., g a G be a basis and g G Q f x, gg i = a ij g j f x mod f. Let χ(t) = det(te (a ij )) be the characteristic polynomial. Then χ is irreducible in Q[t]. = (h/f) x }.

Primary Decomposition p. 2 Splitting over C: Example f = x 2 + y 2

Primary Decomposition p. 2 Splitting over C: Example f = x 2 + y 2 G =< x, y > Q (a i,j ) = ( 0 1/2 1/2 0 )

Primary Decomposition p. 2 Splitting over C: Example f = x 2 + y 2 G =< x, y > Q (a i,j ) = ( 0 1/2 1/2 0 ) χ(t) = t 2 + 1/4

Primary Decomposition p. 2 Splitting over C: Example f = x 2 + y 2 G =< x, y > Q (a i,j ) = ( 0 1/2 1/2 0 ) χ(t) = t 2 + 1/4 gcd(x 2 + y 2, y i 2 2x)gcd(x2 + y 2, y+ i 2 2x) = x2 + y 2

Primary Decomposition p. 2 How to compute the normalization? Let A be a reduced ring, the normalization A is the integral closure of A in the total ring of fractions Q(A).

Primary Decomposition p. 2 How to compute the normalization? Let A be a reduced ring, the normalization A is the integral closure of A in the total ring of fractions Q(A). Let A be a reduced Noetherian ring and J A an ideal containing a non zerodivisor x of A. Then there are natural inclusions of rings A Hom A (J, J) = 1 x (xj : J) A.

Primary Decomposition p. 2 proof For a A, let m a : J J denote the multiplication with a. If m a = 0, then m a (x) = ax = 0 and, hence, a = 0, since x is a non zerodivisor. Thus, a m a defines an inclusion A Hom A (J, J). It is easy to see that for ϕ Hom A (J, J) the element ϕ(x)/x Q(A) is independent of x: for any a J we have ϕ(a) = (1/x) ϕ(xa) = a ϕ(x)/x, since ϕ is A linear. Hence, ϕ ϕ(x)/x defines an inclusion Hom A (J, J) Q(A) mapping x Hom A (J, J) into xj : J = {b A bj xj}. The latter map is also surjective, since any b xj : J defines, via multiplication with b/x, an element ϕ Hom A (J, J) with ϕ(x) = b. Since x is a non zerodivisor, we obtain the isomorphism Hom A (J, J) = (1/x) (xj : J).

Primary Decomposition p. 2 non-normal locus The non normal locus of A is defined as N(A) = {P SpecA A P is not normal}. Let C = Ann A (A/A) = {a A aa A} be the conductor of A in A. Then N(A) = V (C) = {P SpecA P C}.

Primary Decomposition p. 2 Let J A be an ideal containing a non zerodivisor of A. There are natural inclusions of A modules Hom A (J, J) Hom A (J, A) A Hom A (J, J).

Primary Decomposition p. 2 Let J A be an ideal containing a non zerodivisor of A. There are natural inclusions of A modules Hom A (J, J) Hom A (J, A) A Hom A (J, J). If N(A) V (J) then J d A A for some d.

Primary Decomposition p. 2 The embedding of Hom A (J, A) in Q(A) is given by ϕ ϕ(x)/x, where x is a non zerodivisor of J. With this identification we obtain Hom A (J, A) = A : Q(A) J = {h Q(A) hj A} and Hom A (J, J), respectively Hom A (J, J), is identified with those h Q(A) such that hj J, respectively hj J. Then the first inclusion follows. For the second inclusion let h A satisfy hj A. Consider an integral relation h n + a 1 h n 1 + + a n = 0 with a i A. Let g J and multiply the above equation with g n. Then (hg) n + ga 1 (hg) n 1 + + g n a n = 0. Since g J, hg A and, therefore, (hg) n J and hg J. This shows the second inclusion.

Primary Decomposition p. 2 Criterion for Normality Let A be a Noetherian reduced ring and J A an ideal satisfying J contains a non zerodivisor of A, J is a radical ideal, N(A) V (J).

Primary Decomposition p. 2 Criterion for Normality Let A be a Noetherian reduced ring and J A an ideal satisfying J contains a non zerodivisor of A, J is a radical ideal, N(A) V (J). Then A is normal if and only if A = Hom A (J, J).

Primary Decomposition p. 3 proof If A = A then Hom A (J, J) = A. To see the converse, we choose d 0 minimal such that J d A A. If d > 0 then there exists some a J d 1 and h A such that ah A.

Primary Decomposition p. 3 proof If A = A then Hom A (J, J) = A. To see the converse, we choose d 0 minimal such that J d A A. If d > 0 then there exists some a J d 1 and h A such that ah A. But ah A and ah J hj d A, that is, ah Hom A (J, A) A, which is equal to Hom A (J, J), since J = J.

Primary Decomposition p. 3 Let A be a reduced Noetherian ring, let J A be an ideal and x J a non zerodivisor. Then A = Hom A (J, J) if and only if xj : J = x.

Primary Decomposition p. 3 Let A be a reduced Noetherian ring, let J A be an ideal and x J a non zerodivisor. Then A = Hom A (J, J) if and only if xj : J = x. Moreover, let {u 0 = x, u 1,..., u s } be a system of generators for the A module xj : J. Then we can write sx u i u j = xξ ij k u k with suitable ξ ij k A, 1 i j s. k=0 Let (η (k) 0,..., η(k) s ) A s+1, k = 1,..., m, generate syz(u 0,..., u s ), and let I A[t 1,..., t s ] be the ideal ( t 0 := 1) I := *( t i t j sx k=0 ξ ij k t k ) 1 i j s, ( sx ν=0 η ν (k) t ν 1 k m )+, t i u i /x, i = 1,..., s, defines an isomorphism A[t 1,..., t s ]/I = HomA (J, J) = 1 x (xj : J).

Primary Decomposition p. 3 Example Let A := K[x, y]/ x 2 y 3 and J := x, y A. Then x J is a non zerodivisor in A with xj : J = x x, y : x, y = x, y 2, therefore, Hom A (J, J) = 1, y 2 /x. Setting u 0 := x, u 1 := y 2, we obtain u 2 1 = y 4 = x 2 y, that is, ξ 11 0 = y. Hence, we obtain an isomorphism

Primary Decomposition p. 3 normalization(i) Input:I := f 1,..., f k K[x] a prime ideal, x = (x 1,..., x n ). Output: A polynomial ring K[t], t = (t 1,..., t N ), a prime ideal P K[t] and π : K[x] K[t] such that the induced map π : K[x]/I K[t]/P is the normalization of K[x]/I. if I = 0 then return (K[x], 0, id K[x] ); compute r := dim(i); if we know that the singular locus of I is V (x 1,..., x n ) J := x 1,..., x n ; else compute J := the ideal of the (n r) minors of the Jacobian matrix I; J := RADICAL(I + J); choose a J {0}; if aj : J = a return (K[x], I, id K[x] );

Primary Decomposition p. 3 normalization(i) compute a generating system u 0 = a, u 1,..., u s for aj : J; (1) compute a generating system (η 0,..., η(1) s ),..., (η (m) 0,..., η s (m) ) for the module of syzygies syz(u 0,..., u s ) (K[x]/I) s+1 ; compute ξ ij k such that u i u j = P s k=0 a ξij k u k, i, j = 1,... s; change ring to K[x 1,..., x n, t 1,..., t s ], and set (with t 0 := 1) I 1 := {t i t j P s k=0 ξ ij k t k} 0 i j s, { P s ν=0 η ν (k) t ν } 1 k m + IK[x, t]; return NORMALIZATION(I 1 ).

Primary Decomposition p. 3 non-normal locus The ideal Ann A ( HomA (J, J)/A ) A defines the non normal locus. Moreover, Ann A ( HomA (J, J)/A ) = x : (xj : J) for any non zerodivisor x J.

Primary Decomposition p. 3 non normallocus(i) Input: I := f 1,..., f k K[x] a prime ideal, x = (x 1,..., x n ). Output: An ideal N K[x], defining the non normal locus in V (I). If I = 0 then return (K[x]); compute r = dim(i); compute J the ideal of the (n r) minors of the Jacobian matrix of I; J = RADICAL (I + J); choose a J {0}; return ( a : (aj : J) ).