Generalized minimum distance functions and algebraic invariants of Geramita ideals
Susan M. Cooper, Alexandra Seceleanu, Stefan O. Tohaneanu, Maria Vaz Pinto, Rafael H. Villarreal
aa r X i v : . [ m a t h . A C ] S e p GENERALIZED MINIMUM DISTANCE FUNCTIONS AND ALGEBRAICINVARIANTS OF GERAMITA IDEALS
SUSAN M. COOPER, ALEXANDRA SECELEANU, S¸TEFAN O. TOH ˘ANEANU, MARIA VAZ PINTO,AND RAFAEL H. VILLARREAL
Abstract.
Motivated by notions from coding theory, we study the generalized minimum dis-tance (GMD) function δ I ( d, r ) of a graded ideal I in a polynomial ring over an arbitrary fieldusing commutative algebraic methods. It is shown that δ I is non-decreasing as a function of r and non-increasing as a function of d . For vanishing ideals over finite fields, we show that δ I is strictly decreasing as a function of d until it stabilizes. We also study algebraic invariants ofGeramita ideals. Those ideals are graded, unmixed, 1-dimensional and their associated primesare generated by linear forms. We also examine GMD functions of complete intersections andshow some special cases of two conjectures of Toh˘aneanu–Van Tuyl and Eisenbud-Green-Harris. Introduction
Let K be any field, and let C be a linear code that is the image of some K -linear map K s −→ K n . Suppose G is the s × n matrix representing this map with respect to some chosenbases and assume that G has no zero columns. By definition, the minimum (Hamming) distance of C is δ ( C ) := min { wt( v ) | v ∈ C \ { }} , where for any vector w ∈ K n , the weight of w , denoted wt( w ), is the number of nonzero entriesin w . More generally, for 1 ≤ r ≤ dim K ( C ), the r -th generalized Hamming distance , denoted δ r ( C ), is defined as follows. For any subcode, i.e., linear subspace, D ⊆ C define the support of D to be χ ( D ) := { i | there exists ( x , . . . , x n ) ∈ D with x i = 0 } . Then the r -th generalized Hamming distance of C is δ r ( C ) := min D ⊆ C, dim D = r | χ ( D ) | . The weight hierarchy of C is the sequence ( δ ( C ) , . . . , δ k ( C )), where k = dim( C ). Observe that δ ( C ) equals the minimum distance δ ( C ). The study of these weights is related to trellis coding, t –resilient functions, and was motivated by some applications from cryptography [37]. It is thestudy of the generalized Hamming weight of a linear code that motivates our definition of ageneralized minimum distance function for any graded ideal in a polynomial ring [19, 21].If the rank of G is s , then it turns out (see [37]) that(1.1) δ r ( C ) = n − hyp r ( C ) , where hyp r ( C ), is the maximum number of columns of G that span an ( s − r )-dimensionalvector subspace of K s . Moreover, if G also has no proportional columns then the columns of G determine the coordinates of n (projective) points in P s − , not all contained in a hyperplane. Mathematics Subject Classification.
Primary 13P25; Secondary 14G50, 94B27, 11T71.
Key words and phrases. generalized minimum distance, fat points, linear codes, basic parameters, degree,Hilbert function, unmixed ideal, complete intersections, Reed–Muller-type codes.
Denote this set X = { P , . . . , P n } and let I := I ( X ) ⊂ S := K [ t , . . . , t s ] be the defining ideal of X . We have: • the (Krull) dimension of S/I is dim(
S/I ) = 1, and the degree is deg(
S/I ) = n ; • the ideal I is given by I = p ∩ · · · ∩ p n , where p i is the vanishing ideal of the point P i , so I is unmixed, each associated prime ideal p i is generated by linear forms, and I = √ I ; • hyp r ( C ) = max F ∈F r { deg( S/ ( I, F )) } , where F r is the set of r -tuples of linear forms of S thatare linearly independent. With this, we can conclude that δ r ( C ) = deg( S/I ) − max F ∈F r { deg( S/ ( I, F )) } . A similar approach can be taken for projective Reed–Muller-type codes. Let X = { P , . . . , P n } be a finite subset of P s − . Let I := I ( X ) ⊂ S = K [ t , . . . , t s ], be the defining ideal of X . Via arescaling of the homogeneous coordinates of the points P i , we can assume that the first non-zerocoordinate of each P i is 1. Fix a degree d ≥
1. Because of the assumption on the coordinatesof the P i , there is a well-defined K -linear map given by the evaluation of the homogeneouspolynomials of degree d at each point in X . This map is given byev d : S d → K n , f ( f ( P ) , . . . , f ( P n )) , where S d denotes the K -vector space of homogeneous polynomials of S of degree d . The imageof S d under ev d , denoted by C X ( d ), is called a projective Reed-Muller-type code of degree d on X [5, 12, 16]. The parameters of the linear code C X ( d ) are: • length : | X | = deg( S/I ); • dimension : dim K C X ( d ) = H X ( d ), the Hilbert function of S/I in degree d ; • r -th generalized Hamming weight : δ X ( d, r ) := δ r ( C X ( d )).By [14, Theorem 4.5] the r -th generalized Hamming weight of a projective Reed–Muller codeis given by δ X ( d, r ) = deg( S/I ) − max F ∈F d,r { deg( S/ ( I, F ) } , where F d,r the set of r -tuples of forms of degree d in S which are linearly independent over K modulo the ideal I and the maximum is taken to be 0 if F d,r = ∅ .As we can see above, the generalized Hamming weights for any linear code can be interpretedusing the language of commutative algebra. Motivated by the notion of generalized Hammingweight described above and following [14] we define generalized minimum distance (GMD) func-tions for any homogeneous ideal in a polynomial ring. This allows us to extend the notion ofgeneralized Hamming weights to codes arising from algebraic schemes, rather than just fromreduced sets of points. Another advantage to formulating the notion of generalized minimumdistance in the language of commutative algebra is that it allows the use of various homologicalinvariants of graded ideals to study the possible values for these GMD functions.Let S = K [ t , . . . , t s ] = ⊕ ∞ d =0 S d be a polynomial ring over a field K with the standard gradingand let I = (0) be a graded ideal of S . Given d, r ∈ N + , let F d,r be the set: F d,r := { { f , . . . , f r } ⊂ S d | f , . . . , f r are linearly independent over K, ( I : ( f , . . . , f r )) = I } , where f = f + I is the class of f modulo I , and ( I : ( f , . . . , f r )) = { g ∈ S | gf i ∈ I, for all i } .If necessary we denote F d,r by F d,r ( I ). We denote the degree of S/I by deg(
S/I ). LGEBRAIC INVARIANTS OF GERAMITA IDEALS 3
Definition 1.1.
Let I = (0) be a graded ideal of S . The function δ I : N + × N + → Z given by δ I ( d, r ) := ( deg( S/I ) − max { deg( S/ ( I, F )) | F ∈ F d,r } if F d,r = ∅ , deg( S/I ) if F d,r = ∅ , is called the generalized minimum distance function of I , or simply the GMD function of I .This notion recovers (Proposition 3.14) and refines the algebraic-geometric notion of degree.If r = 1 one obtains the minimum distance function of I [24]. In this case we denote δ I ( d, δ I ( d ) and F d,r by F d .The aims of this paper are to study the behavior of δ I , to introduce algebraic methods toestimate this function, and to study the algebraic invariants (minimum distance function, v-number, regularity, socle degrees) of special ideals that we call Geramita ideals. Recall that anideal I ⊂ S is called unmixed if all its associated primes have the same height; this notion issometimes called height unmixed in the literature. We call an ideal I ⊂ S a Geramita ideal if I is an unmixed graded ideal of dimension 1 whose associated primes are generated by linearforms. The defining ideal of the scheme of a finite sets of projective fat points and the unmixedmonomial ideals of dimension 1 are examples of Geramita ideals.The following function is closely related to δ I as illustrated in Eq. (1.1). Definition 1.2.
Let I be a graded ideal of S . The function hyp I : N + × N + → N , given byhyp I ( d, r ) := ( max { deg( S/ ( I, F )) | F ∈ F d,r } if F d,r = ∅ , F d,r = ∅ , is called the hyp function of I .If r = 1, we denote hyp I ( d,
1) by hyp I ( d ). Finding upper bounds for hyp I ( d, r ) is equivalentto finding lower bounds for δ I ( d, r ). If I ( X ) is the vanishing ideal of a finite set X of reducedprojective points, then hyp I ( X ) ( d,
1) is the maximum number of points of X contained in a hyper-surface of degree d (see [33, Remarks 2.7 and 3.4]). There is a similar geometric interpretationfor hyp I ( X ) ( d, r ) [14, Lemma 3.4].To compute δ I ( d, r ) is a difficult problem even when K is a finite field and r = 1. However,we show that a generalized footprint function, which is more computationally tractable, giveslower bounds for δ I ( d, r ). Fix a monomial order ≺ on S . Let in ≺ ( I ) be the initial ideal of I andlet ∆ ≺ ( I ) be the footprint of S/I , consisting of all the standard monomials of S/I with respectto ≺ . The footprint of S/I is also called the
Gr¨obner ´escalier of I . Given integers d, r ≥
1, let M ≺ ,d,r be the set of all subsets M of ∆ ≺ ( I ) d := ∆ ≺ ( I ) ∩ S d with r distinct elements such that(in ≺ ( I ) : ( M )) = in ≺ ( I ). Definition 1.3.
The generalized footprint function of I , denoted fp I , is the function fp I : N + × N + → Z given byfp I ( d, r ) := ( deg( S/I ) − max { deg( S/ (in ≺ ( I ) , M )) | M ∈ M ≺ ,d,r } if M ≺ ,d,r = ∅ , deg( S/I ) if M ≺ ,d,r = ∅ . . If r = 1 one obtains the footprint function of I that was studied in [28] from a theoreticalpoint of view (see [24, 25] for some applications). In this case we denote fp I ( d,
1) simply byfp I ( d ) and M ≺ ,d,r by M ≺ ,d . The importance of the footprint function is that it gives a lowerbound on the generalized minimum degree function (Theorem 3.9) and it is computationally S. M. COOPER, A. SECELEANU, S. O. TOH ˘ANEANU, M. VAZ PINTO, AND R. H. VILLARREAL much easier to determine than the generalized minimum degree function. See the Appendix forscripts that implement these computations.The content of this paper is as follows. In Section 2 we present some of the results andterminology that will be needed throughout the paper. In some of our results we will assume thatthere exists a linear form h that is regular on S/I , that is, ( I : h ) = I . There are wide families ofideals over finite fields that satisfy this hypothesis, e.g., vanishing ideals of parameterized codes[30]. Thus our results can be applied to a variety of Reed–Muller type codes [16], to monomialideals, and to ideals that satisfy | K | > deg( S/ √ I ).In Section 3 we study GMD functions of unmixed graded ideals. The footprint matrix (fp I ( d, r )) and the weight matrix ( δ I ( d, r )) of I are the matrices whose ( d, r )-entries are fp I ( d, r )and δ I ( d, r ), respectively. We show that the entries of each row of the weight matrix form anon-decreasing sequence and that the entries of each column of the weight matrix form a non-increasing sequence (Theorem 3.9). We also show that fp I ( d, r ) is a lower bound for δ I ( d, r )(Theorem 3.9). This was known when I is the vanishing ideal of a finite set of projective points[14, Theorem 4.9].Let I ⊂ S be an unmixed graded ideal whose associated primes are generated by linear forms.In Section 4 we study the minimum distance functions of these ideals. For δ I ( d ) = δ I ( d, regularity index of δ I , denoted reg( δ I ), is the smallest d ≥ δ I ( d ) = 1. If I is prime,we set reg( δ I ) = 1. The regularity index of δ I is the index where the value of this numericalfunction stabilizes (Remark 3.10), named by analogy with the regularity index for the Hilbertfunction of a fat point scheme Z which is the index where the Hilbert function H Z of Z stabilizes.In order to study the behavior of δ I we introduce a numerical invariant called the v- number (Definition 4.1). We give a description for this invariant in Proposition 4.2 that will allow us tocompute it using computer algebra systems, e.g. Macaulay
Proposition 4.6
Let I ( m ⊂ S be an unmixed graded ideal whose associated primes aregenerated by linear forms. Then reg( δ I ) = v( I ) . From the viewpoint of algebraic coding theory it is important to determine reg( δ I ). Indeed let X be a set of projective points over a finite field K , let C X ( d ) be its corresponding Reed-Mullertype code, and let δ X ( d ) be the minimum distance of C X ( d ) (see Section 5), then δ X ( d ) ≥ ≤ d < reg( δ I ( X ) ). Our results give an effective method—that can be applied to anyReed-Muller type code—to compute the regularity index of the minimum distance (Corollary 5.6,Example 4.5).The minimum socle degree s( I ) of S/I (Definition 2.7) was used in [33] to obtain homologicallower bounds for the minimum distance of a fat point scheme Z in P s − . We relate the minimumsocle degree, the v-number and the Castelnuovo-Mumford regularity for Geramita ideals inTheorem 4.10. For radical ideals it is an open problem whether or not reg( δ I ) ≤ reg( S/I )[28, Conjecture 4.2]. In dimension 1, the conjecture is true because of Proposition 4.6 andTheorem 4.10. Moreover, via Theorem 4.10, we can extend the notion of a Cayley–Bacharachscheme [11] by defining the notion of a Cayley–Bacharach ideal (Definition 4.14). It turns outthat Cayley-Bacharach ideals are connected to Reed–Muller type codes and to minimum distancefunctions.Letting H I be the Hilbert function of I , we have δ I ( d ) > deg( S/I ) − H I ( d ) + 1 for some d ≥ I is unmixed of dimension at least 2 (Proposition 4.21). One of our main results is: Theorem 4.19 If I ⊂ S is a Geramita ideal and there exists h ∈ S regular on S/I , then δ I ( d ) ≤ deg( S/I ) − H I ( d ) + 1 LGEBRAIC INVARIANTS OF GERAMITA IDEALS 5 for d ≥ or equivalently H I ( d ) − ≤ hyp I ( d ) for d ≥ . This inequality is well known when I is the vanishing ideal of a finite set of projective points[30, p. 82]. In this case the inequality is called the Singleton bound [34, Corollary 1.1.65].Projective Reed–Muller-type codes are studied in Section 5.The main result of Section 5 shows that the entries of each column of the weight matrix( δ X ( d, r )) form a decreasing sequence until they stabilize.In particular one recovers the case when X is a set, lying on a projective torus, parameterizedby a finite set of monomials [13, Theorem 12]. Then we show that δ X ( d, H X ( d )) is equal to | X | for d ≥ Conjecture 6.3
Let X be a finite in P s − and suppose that I = I ( X ) is a complete intersectiongenerated by f , . . . , f c , c = s − , with d i = deg( f i ) , and ≤ d i ≤ d i +1 for all i . (a) (Toh˘aneanu–Van Tuyl [33, Conjecture 4.9]) δ I (1) ≥ ( d − d · · · d c . (b) (Eisenbud-Green-Harris [8, Conjecture CB10]) If f , . . . , f c are quadratic forms, then hyp I ( d ) ≤ c − c − d for ≤ d ≤ c or equivalently δ I ( d ) ≥ c − d for ≤ d ≤ c . We prove part (a) of this conjecture, in a more general setting, when I is equigenerated, thatis, all minimal homogeneous generators have the same degree (Proposition 6.4, Remark 6.6).The conjecture also holds for P [33, Theorem 4.10] (Corollary 6.5). According to [8], part (b)of this conjecture is true for the following values of d : 1 , c − , c .For all unexplained terminology and additional information we refer to [4, 6, 27] (for thetheory of Gr¨obner bases, commutative algebra, and Hilbert functions), and [23, 34] (for thetheory of error-correcting codes and linear codes).2. Preliminaries
In this section we present some of the results that will be needed throughout the paper andintroduce some more notation. All results of this section are well-known. To avoid repetitions,we continue to employ the notations and definitions used in Section 1.
Commutative algebra.
Let I = (0) be a graded ideal of S of Krull dimension k . The Hilbertfunction of S/I is: H I ( d ) := dim K ( S d /I d ) for d = 0 , , , . . . , where I d = I ∩ S d . By a theoremof Hilbert [32, p. 58], there is a unique polynomial P I ( x ) ∈ Q [ x ] of degree k − H I ( d ) = P I ( d ) for d ≫
0. By convention the degree of the zero polynomial is − degree or multiplicity of S/I is the positive integerdeg(
S/I ) := ( ( k − d →∞ H I ( d ) /d k − if k ≥ , dim K ( S/I ) if k = 0 . As usual ht( I ) will denote the height of the ideal I . By the dimension of I (resp. S/I ) we meanthe Krull dimension of
S/I denoted by dim(
S/I ).One of the most useful and well-known facts about the degree is its additivity:
Proposition 2.1. (Additivity of the degree [29, Proposition 2.5]) If I is an ideal of S and I = q ∩ · · · ∩ q m is an irredundant primary decomposition, then deg( S/I ) = X ht( q i )=ht( I ) deg( S/ q i ) . S. M. COOPER, A. SECELEANU, S. O. TOH ˘ANEANU, M. VAZ PINTO, AND R. H. VILLARREAL If F ⊂ S , the ideal quotient of I with respect to ( F ) is given by ( I : ( F )) = { h ∈ S | hF ⊂ I } .An element f of S is called a zero-divisor of S/I —as an S -module—if there is 0 = a ∈ S/I suchthat f a = 0, and f is called regular on S/I if f is not a zero-divisor. Thus f is a zero-divisor ifand only if ( I : f ) = I . An associated prime of I is a prime ideal p of S of the form p = ( I : f )for some f in S . Theorem 2.2. [36, Lemma 2.1.19, Corollary 2.1.30] If I is an ideal of S and I = q ∩ · · · ∩ q m is an irredundant primary decomposition with rad( q i ) = p i , then the set of zero-divisors Z ( S/I ) of S/I is equal to S mi =1 p i , and p , . . . , p m are the associated primes of I . Definition 2.3. If I is a graded ideal of S , the Hilbert series of S/I , denoted F I ( x ), is given by F I ( x ) = ∞ X d =0 H I ( d ) x d , where x is a variable . Theorem 2.4. (Hilbert–Serre [32, p. 58])
Let I ⊂ S be a graded ideal of dimension k . Thenthere is a unique polynomial h ( x ) ∈ Z [ x ] such that F I ( x ) = h ( x )(1 − x ) k and h (1) > . Remark 2.5.
The leading coefficient of the Hilbert polynomial P I ( x ) is equal to h (1) / ( k − h (1) is equal to deg( S/I ). Definition 2.6.
Let I ⊂ S be a graded ideal. The a -invariant of S/I , denoted a ( S/I ), is thedegree of F I ( x ) as a rational function, that is, a ( S/I ) = deg( h ( x )) − k . If h ( x ) = P ri =0 h i x i , h i ∈ Z , h r = 0, the vector ( h , . . . , h r ) is called the h -vector of S/I . Definition 2.7.
Let I ⊂ S be a graded ideal and let F be the minimal graded free resolutionof S/I as an S -module: F : 0 → M j S ( − j ) b g,j → · · · → M j S ( − j ) b ,j → S → S/I → . The
Castelnuovo–Mumford regularity of S/I ( regularity of S/I for short) and the minimum socledegree (s- number for short) of
S/I are defined asreg(
S/I ) = max { j − i | b i,j = 0 } and s( I ) = min { j − g | b g,j = 0 } . If S/I is Cohen-Macaulay (i.e. g = dim( S ) − dim( S/I )) and there is a unique j such that b g,j = 0, then the ring S/I is called level . In particular, a level ring for which the unique j suchthat b g,j = 0 is b g,j = 1 is called Gorenstein .An excellent reference for the regularity of graded ideals is the book of Eisenbud [7].
Definition 2.8.
The regularity index of the Hilbert function of
S/I , or simply the regularityindex of S/I , denoted ri(
S/I ), is the least integer n ≥ H I ( d ) = P I ( d ) for d ≥ n .The next result is valid over any field; see for instance [36, Theorem 5.6.4]. Theorem 2.9. [11]
Let I be a graded ideal with depth( S/I ) > . The following hold. (i) If dim( S/I ) ≥ , then H I ( i ) < H I ( i + 1) for i ≥ . (ii) If dim( S/I ) = 1 , then there is an integer r and a constant c such that H I (0) < H I (1) < · · · < H I ( r − < H I ( i ) = c for i ≥ r. Lemma 2.10.
Let I ⊂ J ⊂ S be graded ideals of the same height. The following hold. LGEBRAIC INVARIANTS OF GERAMITA IDEALS 7 (a) [9, Lemma 8] If I and J are unmixed, then I = J if and only if deg( S/I ) = deg(
S/J ) . (b) If I ( J , then deg( S/I ) > deg( S/J ) .Proof. (b) Since any associated prime of J/I is an associated prime of
S/I , dim(
J/I ) = dim(
S/I ).From the short exact sequence 0 → J/I → S/I → S/J → S/I ) = deg(
J/I ) + deg(
S/J ). As
J/I is not zero, one has deg(
S/I ) > deg( S/J ). (cid:3) Lemma 2.11. [36, p. 122]
Let I ⊂ S a graded ideal of height r . If K is infinite and I isminimally generated by forms of degree p ≥ , then there are forms f , . . . , f m of degree p in I such that f , . . . , f r is a regular sequence and I is minimally generated by f , . . . , f m . The footprint of an ideal.
Let ≺ be a monomial order on S and let (0) = I ⊂ S be an ideal.If f is a non-zero polynomial in S , the leading monomial of f is denoted by in ≺ ( f ). The initialideal of I , denoted by in ≺ ( I ), is the monomial ideal given by in ≺ ( I ) = ( { in ≺ ( f ) | f ∈ I } ).We will use the following multi-index notation: for a = ( a , . . . , a s ) ∈ N s , set t a := t a · · · t a s s .A monomial t a is called a standard monomial of S/I , with respect to ≺ , if t a is not in the idealin ≺ ( I ). A polynomial f is called standard if f = 0 and f is a K -linear combination of standardmonomials. The set of standard monomials, denoted ∆ ≺ ( I ), is called the footprint of S/I . Theimage of the standard polynomials of degree d , under the canonical map S S/I , x x , isequal to S d /I d , and the image of ∆ ≺ ( I ) is a basis of S/I as a K -vector space. This is a classicalresult of Macaulay (for a modern approach see [4, Chapter 5]). In particular, if I is graded, then H I ( d ) is the number of standard monomials of degree d . Lemma 2.12. [3, p. 3]
Let I ⊂ S be an ideal generated by G = { g , . . . , g r } , then ∆ ≺ ( I ) ⊂ ∆ ≺ (in ≺ ( g ) , . . . , in ≺ ( g r )) . Lemma 2.13. [14, Lemma 4.7]
Let ≺ be a monomial order, let I ⊂ S be an ideal, let F = { f , . . . , f r } be a set of polynomial of S of positive degree, and let in ≺ ( F ) = { in ≺ ( f ) , . . . , in ≺ ( f r ) } be the set of initial terms of F . If (in ≺ ( I ) : (in ≺ ( F ))) = in ≺ ( I ) , then ( I : ( F )) = I . Let ≺ be a monomial order and let F ≺ ,d,r be the set of all subsets F = { f , . . . , f r } of S d suchthat ( I : ( F )) = I , f i is a standard polynomial for all i , f , . . . , f r are linearly independent overthe field K , and in ≺ ( f ) , . . . , in ≺ ( f r ) are distinct monomials.The next result is useful for computations with Macaulay
Proposition 2.14. [14, Proposition 4.8]
The generalized minimum distance function of I isgiven by the following formula δ I ( d, r ) = (cid:26) deg( S/I ) − max { deg( S/ ( I, F )) | F ∈ F ≺ ,d,r } if F ≺ ,d,r = ∅ , deg( S/I ) if F ≺ ,d,r = ∅ . An ideal I ⊂ S is called radical if I is equal to its radical. The radical of I is denoted by √ I . Lemma 2.15. [14, Lemma 3.3]
Let I ⊂ S be a radical unmixed graded ideal. If F = { f , . . . , f r } is a set of homogeneous polynomials of S \ { } , ( I : ( F )) = I , and A is the set of all associatedprimes of S/I that contain F , then ht( I ) = ht( I, F ) , A 6 = ∅ , and deg( S/ ( I, F )) = X p ∈A deg( S/ p ) . S. M. COOPER, A. SECELEANU, S. O. TOH ˘ANEANU, M. VAZ PINTO, AND R. H. VILLARREAL Generalized minimum distance function of a graded ideal
In this section we study the generalized minimum distance function of a graded ideal.Part (c) of the next lemma was known for vanishing ideals and part (b) for unmixed radicalideals [14, Proposition 3.5, Lemma 4.1].
Lemma 3.1.
Let I ⊂ S be an unmixed graded ideal, let ≺ be a monomial order, and let F be afinite set of homogeneous polynomials of S such that ( I : ( F )) = I . The following hold. (a) ht( I ) = ht( I, F )(b) deg( S/ ( I, F )) < deg( S/I ) if I is an unmixed ideal and ( F ) I . (c) deg( S/I ) = deg( S/ ( I : ( F ))) + deg( S/ ( I, F )) if I is an unmixed radical ideal. (d) [14, Lemma 4.1] deg( S/ ( I, F )) ≤ deg( S/ (in ≺ ( I ) , in ≺ ( F ))) ≤ deg( S/I ) .Proof. (a) As I ( ( I : ( F )), there is g ∈ S \ I such that g ( F ) ⊂ I . Hence the ideal ( F ) iscontained in the set of zero-divisors of S/I . Thus, by Theorem 2.2 and since I is unmixed, ( F )is contained in an associated prime ideal p of S/I of height ht( I ). Thus I ⊂ ( I, F ) ⊂ p , andconsequently ht( I ) = ht( I, F ). Therefore the set of associated primes of (
I, F ) of height equalto ht( I ) is not empty and is equal to the set of associated primes of S/I that contain ( F ).(b) The inequality follows from part (a) and Lemma 2.10 (b).(c) Let p , . . . , p m be the associated primes of S/I . As I is a radical ideal, one has thedecompositions I = m \ i =1 p i and ( I : ( F )) = m \ i =1 ( p i : ( F )) . Note that ( p i : ( F )) = S if F ⊂ p i and ( p i : ( F )) = p i if F p i . Therefore, using the additivityof the degree of Proposition 2.1 and Lemma 2.15, we getdeg( S/ ( I : ( F ))) = X F p i deg( S/ p i ) and deg( S/ ( I, F )) = X F ⊂ p i deg( S/ p i ) . Thus deg(
S/I ) = P mi =1 deg( S/ p i ) = deg( S/ ( I : ( F ))) + deg( S/ ( I, F )). (cid:3)
Definition 3.2.
Let I ⊂ S be a graded ideal. A sequence f , . . . , f r of elements of S is called a( d, r )- sequence of S/I if the set F = { f , . . . , f r } is in F d,r Lemma 3.3.
Let I ⊂ S be a graded ideal. A sequence f , . . . , f r is a ( d, r ) -sequence of S/I ifand only if the following conditions hold (a) f , . . . , f r are homogeneous polynomials of S of degree d ≥ , (b) ( I : ( f , . . . , f r )) = I , and (c) f i / ∈ ( I, f , . . . , f i − ) for i = 1 , . . . , r , where we set f = 0 .Proof. The proof is straightforward. (cid:3)
Definition 3.4. If I ⊂ S is a graded ideal, the Vasconcelos function of I is the function ϑ I : N + × N + → N given by ϑ I ( d, r ) := (cid:26) min { deg( S/ ( I : ( F ))) | F ∈ F d,r } if F d,r = ∅ , deg( S/I ) if F d,r = ∅ . The next result was shown in [14, Theorem 4.5] for vanishing ideals over finite fields.
LGEBRAIC INVARIANTS OF GERAMITA IDEALS 9
Theorem 3.5.
Let I ⊂ S be a graded unmixed radical ideal. Then ϑ I ( d, r ) = δ I ( d, r ) for d ≥ and ≤ r ≤ H I ( d ) . Proof. If F d,r = ∅ , then δ I ( d, r ) and ϑ I ( d, r ) are equal to deg( S/I ). Now assume that F d,r = ∅ .Using Lemma 3.1(c), we obtain ϑ I ( d, r ) = min { deg( S/ ( I : ( F ))) | F ∈ F d,r } = min { deg( S/I ) − deg( S/ ( I, F )) | F ∈ F d,r } = deg( S/I ) − max { deg( S/ ( I, F )) | F ∈ F d,r } = δ I ( d, r ) . ✷ As the next result shows for r = 1 we do not need the assumption that I is a radical ideal.For r ≥ Example 3.6.
Let I be the ideal ( t , t t , t ) of the polynomial ring S = K [ t , t ] over a field K and let F = { t , t } . Then ( I : ( F )) = ( I, F ) = ( t , t ) and3 = deg( S/I ) = deg( S/ ( I : ( F ))) + deg( S/ ( I, F )) = 2 . Theorem 3.7. [24, Theorem 4.4]
Let I ⊂ S be an unmixed graded ideal. If m = ( t , . . . , t s ) and d ≥ is an integer such that m d I , then δ I ( d ) = min { deg( S/ ( I : f )) | f ∈ S d \ I } . Recall from the introduction that the definition of δ I ( d, r ) was motivated by the notion ofgeneralized Hamming weight of a linear code [19, 37]. The following compilation of facts reflectsthe monotonicity of the generalized minimum distance function with respect of its two inputvalues for the case of linear codes corresponding to reduced sets of points. Theorem 3.8.
Let C be a linear code of length m and dimension k . The following hold. (a) [37, Theorem 1, Corollary 1] 1 ≤ δ ( C ) < · · · < δ k ( C ) ≤ m . (b) [34, Corollary 1.1.65] r ≤ δ r ( C ) ≤ m − k + r for r = 1 , . . . , k . (c) If δ ( C ) = m − k + 1 , then δ r ( C ) = m − k + r for r = 1 , . . . , k .Proof. (c): By (a), one has m − k + 1 = δ ( C ) ≤ δ i ( C ) − ( r − m − k + r ≤ δ i ( C ) and,by (b), equality holds. (cid:3) Below we consider more generally the behavior of the generalized minimum distance functionand the footprint function for arbitrary graded ideals. The next result shows that the entries ofany row (resp. column) of the weight matrix of I form a non-decreasing (resp. non-increasing)sequence. Parts (a)-(c) of the next result are broad generalizations of [14, Theorem 4.9] and [28,Theorem 3.6]. Theorem 3.9.
Let I ⊂ S be an unmixed graded ideal, let ≺ be a monomial order on S , and let d ≥ , r ≥ be integers. The following hold. (a) fp I ( d, r ) ≤ δ I ( d, r ) for ≤ r ≤ H I ( d ) . (b) δ I ( d, r ) ≥ . (c) fp I ( d, r ) ≥ if in ≺ ( I ) is unmixed. (d) δ I ( d, r ) ≤ δ I ( d, r + 1) . (e) If there is h ∈ S regular on S/I , then δ I ( d, r ) ≥ δ I ( d + 1 , r ) ≥ . Proof. (a) If F d,r = ∅ , then δ I ( d, r ) = deg( S/I ) ≥ fp I ( d, r ). Now assume F d,r = ∅ . Let F beany set in F ≺ ,d,r . By Lemma 2.13, in ≺ ( F ) is in M ≺ ,d,r , and by Lemma 3.1, deg( S/ ( I, F )) ≤ deg( S/ (in ≺ ( I ) , in ≺ ( F ))). Hence, by Proposition 2.14 and Lemma 3.1(b), fp I ( d, r ) ≤ δ I ( d, r ).(b) If F d,r = ∅ , then δ I ( d, r ) = deg( S/I ) ≥
1, and if F d,r = ∅ , then using Lemma 3.1(b) itfollows that δ I ( d, r ) ≥ M ≺ ,d,r = ∅ , then fp I ( d, r ) = deg( S/I ) ≥
1. Next assume that M ≺ ,d,r is not empty andpick M in M ≺ ,d,r such thatfp I ( d, r ) = deg( S/I ) − deg( S/ (in ≺ ( I ) , M )) . As in ≺ ( I ) is unmixed, by Lemma 3.1(b), fp I ( d, r ) ≥ F d,r +1 is empty, then δ I ( d, r ) ≤ deg( S/I ) = δ I ( d, r + 1). We may then assume F d,r +1 is not empty and pick F = { f , . . . , f r +1 } in F d,r +1 such that hyp I ( d, r + 1) = deg( S/ ( I, F )).Setting F ′ = { f , . . . , f r } and noticing that I ( ( I : ( F )) ⊂ ( I : ( F ′ )), we get F ′ ∈ F d,r . By theproof of Lemma 3.1, one has ht( I ) = ht( I, F ) = ht(
I, F ′ ). Taking Hilbert functions in the exactsequence 0 −→ ( I, F ) / ( I, F ′ ) −→ S/ ( I, F ′ ) −→ S/ ( I, F ) −→ S/ ( I, F ′ )) ≥ deg( S/ ( I, F )). Thereforehyp I ( d, r ) ≥ deg( S/ ( I, F ′ )) ≥ deg( S/ ( I, F )) = hyp I ( d, r + 1) ⇒ δ I ( d, r ) ≤ δ I ( d, r + 1) . (e) By part (b), δ I ( d, r ) ≥ d ≥
1. Assume F d,r = ∅ . Then δ I ( d, r ) = deg( S/I ). If the set F d +1 ,r is empty, one has δ I ( d, r ) = δ I ( d + 1 , r ) = deg( S/I ) . If the set F d +1 ,r is not empty, there is F ∈ F d +1 ,r such that δ I ( d + 1 , r ) = deg( S/I ) − deg( S/ ( I, F )) ≤ deg( S/I ) = δ I ( d, r ) . Thus we may now assume F d,r = ∅ . Pick F = { f , . . . , f r } in F d,r such that δ I ( d, r ) = deg( S/I ) − deg( S/ ( I, F )) . By assumption there exists h ∈ S such that ( I : h ) = I . Hence the set hF = { hf i } ri =1 islinearly independent over K , hF ⊂ S d +1 , and I ( ( I : F ) ⊂ ( I : hF ) , that is, hF is in F d +1 ,r . Note that there exists p ∈ Ass(
S/I ) that contains (
I, F ) (see Lemma 3.1(a)).Hence the ideals (
I, F ) and (
I, hF ) have the same height because a prime ideal p ∈ Ass(
S/I )contains (
I, F ) if and only if p contains ( I, hF ). Therefore taking Hilbert functions in the exactsequence 0 −→ ( I, F ) / ( I, hF ) −→ S/ ( I, hF ) −→ S/ ( I, F ) −→ S/ ( I, hF )) ≥ deg( S/ ( I, F )). As a consequence we get δ I ( d, r ) = deg( S/I ) − deg( S/ ( I, F )) ≥ deg( S/I ) − deg( S/ ( I, hF )) ≥ deg( S/I ) − max { deg( S/ ( I, F ′ )) | F ′ ∈ F d +1 ,r } = δ I ( d + 1 , r ) . ✷ Remark 3.10. (a) Let I be a non-prime ideal and let p be an associated prime of I . There is f ∈ S d , d ≥
1, such that ( I : f ) = p . Note that f ∈ F d . By Theorem 3.7 one has δ I ( d ) = 1.(b) If dim( S/I ) ≥
1, then reg( δ I ) is the smallest n ≥ δ I ( d ) = 1 for d ≥ n . Thisfollows from Theorems 3.7 and 3.9. LGEBRAIC INVARIANTS OF GERAMITA IDEALS 11
Example 3.11.
Let S = K [ t , . . . , t ] be a polynomial ring over the finite field K = F andlet I be the ideal ( t t − t t , t t − t t ). The regularity and the degree of S/I are 2 and 4,respectively, and H I (1) = 6, H I (2) = 19. Using Procedure A.2 and Theorem 3.9(a) we obtain:(fp I ( d, r )) = (cid:20) ∞ (cid:21) , d = 1 , r = 1 , . . . , , and ( δ I (1 , , . . . , δ I (1 , , , , , Definition 3.12.
If fp I ( d ) = δ I ( d ) for d ≥
1, we say that I is a Geil–Carvalho ideal . Iffp I ( d, r ) = δ I ( d, r ) for d ≥ r ≥
1, we say that I is a strongly Geil–Carvalho ideal .The next result generalizes [25, Proposition 3.11]. Proposition 3.13. If I is an unmixed monomial ideal and ≺ is any monomial order, then δ I ( d, r ) = fp I ( d, r ) for d ≥ and r ≥ , that is, I is a strongly Geil–Carvalho ideal.Proof. The inequality δ I ( d, r ) ≥ fp I ( d, r ) follows from Theorem 3.9(a). To show the reverseinequality notice that M ≺ ,d,r ⊂ F ≺ ,d,r because one has I = in ≺ ( I ). Also notice that M ≺ ,d,r = ∅ if and only if F ≺ ,d,r = ∅ , this follows from the proof of [14, Proposition 4.8]. Therefore one hasfp I ( d, r ) ≥ δ I ( d, r ). (cid:3) Proposition 3.14. If I ⊂ S is an unmixed graded ideal and dim( S/I ) ≥ , then δ I ( d, H I ( d )) = deg( S/I ) for d ≥ . Proof.
We set r = H I ( d ). It suffices to show that F d,r = ∅ . We proceed by contradiction.Assume that F d,r is not empty and let F = { f , . . . , f r } be an element of F d,r . Let p , . . . , p m bethe associated primes of I . As I ( ( I : ( F )), we can pick g ∈ S such that g ( F ) ⊂ I and g / ∈ I .Then ( F ) is contained ∪ mi =1 p i , and consequently ( F ) ⊂ p i for some i . Since r = H I ( d ), one has S d /I d = Kf ⊕ · · · ⊕ Kf r ⇒ S d = Kf + · · · + Kf r + I d . Hence S d ⊂ p i , that is, m d ⊂ p i , where m = ( t , . . . , t s ). Therefore p i = m , a contradictionbecause I is unmixed and dim( S/I ) ≥ (cid:3) Example 3.15.
Let S = K [ t , t , t ] be a polynomial ring over a field K and let (fp I ( d, r ))and ( δ I ( d, r )) be the footprint matrix and the weight matrix of the ideal I = ( t , t t ). Theregularity and the degree of S/I are 3 and 6. Using Procedure A.1 we obtain:(fp I ( d, r )) = ∞ ∞ ∞ ∞ . If r > H I ( d ), then M ≺ ,d,r = ∅ and the ( d, r )-entry of this matrix is equal to 6, but in thiscase we write ∞ for computational reasons. Therefore, by Proposition 3.13, (fp I ( d, r )) is equalto ( δ I ( d, r )). Setting F = { t t , t t , t t , t t } and F ′ = { t t , t t , t t + t , t t } , we get δ I (3 ,
4) = deg(
S/I ) − deg( S/ ( I, F ) = 4 and deg(
S/I ) − deg( S/ ( I, F ′ ) = 5 . Thus δ I (3 ,
4) is attained at F .4. Minimum distance function of a graded ideal
In this section we study minimum distance functions of unmixed graded ideals whose associ-ated primes are generated by linear forms and the algebraic invariants of Geramita ideals.
Minimum distance function for unmixed ideals.
We begin by introducing the fol-lowing numerical invariant which will be used to express the regularity index of the minimumdistance function (Proposition 4.6).
Definition 4.1.
The v- number of a graded ideal I , denoted v( I ), is given byv( I ) := ( min { d ≥ | there exists f ∈ S d and p ∈ Ass( I ) with ( I : f ) = p } if I ( m , I = m , where Ass( I ) is the set of associated primes of S/I and m = ( t , . . . , t s ) is the irrelevant maximalideal of S .The v -number is finite for any graded ideal by the definition of associated primes. If p is aprime ideal and p = m , then v( p ) = 1.Let I ( m ⊂ S be a graded ideal and let p , . . . , p m be its associated primes. One can definethe v-number of I locally at each p i byv p i ( I ) := min { d ≥ | ∃ f ∈ S d with ( I : f ) = p i } . The v-number of I is equal to min { v p ( I ) , . . . , v p m ( I ) } . If I = I ( X ) is the vanishing ideal of afinite set X = { P , . . . , P m } of reduced projective points and p i is the vanishing ideal of P i , thenv p i ( I ) is the degree of P i in X in the sense of [11, Definition 2.1].We give an alternate description for the v-number using initial degrees of certain modules.This will allow us to compute the v-number using Macaulay M = 0 we denote α ( M ) = min { deg( f ) | f ∈ M, f = 0 } . By convention, for M = 0 we set α (0) = 0. Proposition 4.2.
Let I ⊂ S be an unmixed graded ideal. Then I ( ( I : p ) for p ∈ Ass( I ) , v( I ) = min { α (( I : p ) /I ) | p ∈ Ass( I ) } , and α (( I : p ) /I ) = v p ( I ) for p ∈ Ass( I ) .Proof. The strict inclusion I ( ( I : p ) follows from the equivalence of Eq. (4.1) below. As apreliminary step of the proof of the equality we establish that for a prime p ∈ Ass( I ) we have(4.1) ( I : f ) = p if and only if f ∈ ( I : p ) \ I. If ( I : f ) = p , it is clear that we have f ∈ ( I : p ) and since ( I : f ) = S it follows that f / ∈ I .Conversely, if f ∈ ( I : p ) \ I , then p ⊂ ( I : f ). Let q ∈ Ass( I : f ), which is a nonempty setsince f / ∈ I . Since Ass( I : f ) ⊂ Ass( I ) and I is height unmixed, we have ht( q ) = ht( p ) and p ⊂ ( I : f ) ⊂ q . It follows that p = ( I : f ) = q .The equivalence of Eq. (4.1) implies that α (( I : p ) /I ) = v p ( I ), and shows the equality { f | ( I : f ) = p for some p ∈ Ass( I ) } = [ p ∈ Ass( I ) ( I : p ) \ I. The claim now follows by considering the minimum degree of a homogeneous element in theabove sets. (cid:3)
Example 4.3.
Let S = Q [ t , t , t , t ] be a polynomial ring over the rational numbers and let I be the ideal of S given by I = ( t , t , t , t t t ) ∩ ( t , t , t , t t t ) ∩ ( t , t , t ) ∩ ( t , t , t ) . The associated primes of I are p = ( t , t , t ) , p = ( t , t , t ) , p = ( t , t , t ) , p = ( t , t , t ).Using Proposition 4.2 together with Procedure A.3 we get s( I ) = 10, v( I ) = 12, reg( S/I ) = 19,
LGEBRAIC INVARIANTS OF GERAMITA IDEALS 13 v p ( I ) = 12, v p ( I ) = 15, v p i ( I ) = 18 for i = 3 ,
4. Thus the minimum socle degree s( I ) can besmaller than the v-number v( I ). Corollary 4.4. If I ( m is a graded ideal of S and dim( S/I ) = 0 , then the minimum socledegree s( I ) := α (( I : m ) /I ) of S/I is equal to v( I ) .Proof. The socle of
S/I is given by Soc(
S/I ) = ( I : m ) /I . Thus, by Proposition 4.2, one has theequality s( I ) = v( I ). (cid:3) This corollary does not hold in dimension 1. There are examples of Geramita monomial idealssatisfying the strict inequality s( I ) < v( I ) (see Example 4.3). If S/I is a Cohen–Macaulay ring,the socle is understood to be the socle of some Artinian reduction of
S/I by linear forms.
Example 4.5.
Let K be the finite field F and let X be the following set of points in P :[(1 , , , [(1 , , , [(1 , , , [(1 , , , [(1 , , , [(1 , , , [(0 , , , [(0 , , , [(0 , , , [(0 , , . Using Propositions 4.2 and 4.6, together with Procedure A.4, we get v( I ) = reg( δ X ) = 3,reg( S/I ) = 4, δ X (1) = 6, δ X (2) = 3, and δ X ( d ) = 1 for d ≥
3. The vanishing ideal of X isgenerated by t t − t t , t t − t t , and t t − t t . Proposition 4.6.
Let I ( m ⊂ S be an unmixed graded ideal whose associated primes aregenerated by linear forms. Then reg( δ I ) = v( I ) .Proof. Let p , . . . , p m be the associated primes of I . We may assume that I not a prime ideal,otherwise reg( δ I ) = v( I ) = 1. If d = v( I ), there are f ∈ S d and p i such that ( I : f ) = p i .Then, by Theorem 3.7, one has δ I ( d ) = 1. Thus reg( δ I ) ≤ v( I ).To show the reverse inequality set we set d = reg( δ I ). Then δ I ( d ) = 1. Note that m d I ;otherwise F d ( I ) = ∅ and by definition δ I ( d ) is equal to deg( S/I ), a contradiction because I ( m and by Lemma 2.10 deg( S/I ) >
1. Then, by Theorem 3.7, there is f ∈ S d \ I suchthat δ I ( d ) = deg( S/ ( I : f )) = 1. Let I = ∩ mi =1 q i be the minimal primary decomposition of I ,where q i is a p i -primary ideal. Note that ( q i : f ) is a primary ideal if f / ∈ q i because S/ ( q i : f ) isembedded in S/ q i . Thus the primary decomposition of ( I : f ) is ∩ f / ∈ q i ( q i : f ). Therefore, by theadditivity of the degree of Proposition 2.1, we get that ( I : f ) = ( q k : f ) for some k such that f / ∈ q k and deg( S/ ( q k : f )) = 1. Since S/ p k has also degree 1 and ( q k : f ) ⊂ p k , by Lemma 2.10,we get ( I : f ) = ( q k : f ) = p k , and consequently v( I ) ≤ reg( δ I ). (cid:3) Corollary 4.7.
Let I ⊂ S be an unmixed radical graded ideal. If all the associated primes of I are generated by linear forms and v = v( I ) is its v -number, then δ I (1) > · · · > δ I ( v − > δ I ( v ) = δ I ( d ) = 1 for d ≥ v. Proof.
It follows from [28, Theorem 3.8] and Proposition 4.6. (cid:3)
The minimum distance function behaves well asymptotically.
Corollary 4.8.
Let I ( m ⊂ S be an unmixed graded ideal of dimension ≥ whose associatedprimes are generated by linear forms. Then δ I ( d ) = 1 for d ≥ v( I ) .Proof. This follows from Remark 3.10(b) and Proposition 4.6. (cid:3)
The next result relates the minimum socle degree and the v-number.
Proposition 4.9.
Let I ⊂ S be an unmixed non-prime graded ideal whose associated primesare generated by linear forms and let h ∈ S be a regular element on S/I . The following hold: (a) If δ I ( d ) = deg( S/ ( I : f )) , f ∈ F d ∩ ( I, h ) , then d ≥ and δ I ( d ) = δ I ( d − . (b) If S/I is Cohen–Macaulay, then v( I, h ) ≤ v( I ) . (c) If K is infinite and S/I is Cohen–Macaulay, then s( I ) ≤ v( I ) .Proof. (a) Writing f = g + f h , for some g ∈ I d and f ∈ S d − , one has ( I : f ) = ( I : f ). Notethat d ≥
2, otherwise if d = 1, then ( I : f ) = I , a contradiction because f ∈ F d . Thereforenoticing that f ∈ F d − , by Theorems 3.7 and 3.9, we obtain δ I ( d ) = deg( S/ ( I : f )) = deg( S/ ( I : f )) ≥ δ I ( d − ≥ δ I ( d ) ⇒ δ I ( d ) = δ I ( d − . (b) We set v = v( I ). By Proposition 4.2 there is an associated prime p of I and f ∈ ( I : p ) \ I such that f ∈ S v . Then ( I : f ) = p , f ∈ F d , and δ I ( d ) = deg( S/ ( I : f )) = 1. We claim that f is not in ( I, h ). If f ∈ ( I, h ), then by part (a) one has v ≥ δ I ( v −
1) = 1, a contradictionbecause v is the regularity index of δ I (see Proposition 4.6). Thus f / ∈ ( I, h ). Next we show theequality ( p , h ) = (( I, h ) : f ). The inclusion “ ⊂ ” is clear because ( I : f ) = p . Take an associatedprime p ′ of (( I, h ) : f ). The height of p ′ is ht( I ) + 1 because ( I, h ) is Cohen–Macaulay. Then p ′ = ( p ′′ , h ) for some p ′′ in Ass( I ). Taking into account that p and p ′′ are generated by linearforms, we get the equality ( p , h ) = ( p ′′ , h ). Thus ( p , h ) is equal to (( I, h ) : f ). Hence δ ( I,h ) ( v ) = 1,and consequently v( I, h ) = reg( δ ( I,h ) ) ≤ reg( δ I ) = v( I ) = v .(c) There exists a system of parameters h = h , . . . , h t of S/I consisting of linear forms, where t = dim( S/I ). As
S/I is Cohen–Macaulay, h is a regular sequence on S/I . Hence, by part (b),we obtain v(
I, h ) = v(
I, h , . . . , h t ) ≤ · · · ≤ v( I, h ) ≤ v( I ) . Thus, by Corollary 4.4, we get s( I ) = s( I, h ) = α ((( I, h ) : m ) / ( I, h )) = v(
I, h ) ≤ v( I ). (cid:3) Minimum distance function for Geramita ideals and Cayley-Bacharach ideals.
The minimum socle degree s( I ), the local v-number v p ( I ), and the regularity reg( S/I ), arerelated below. For complete intersections of dimension 1 they are all equal. In particular in thiscase one has δ I ( d ) ≥ ≤ d < reg( S/I ). Theorem 4.10.
Let I ⊂ S be a Geramita ideal and p ∈ Ass( I ) . If I is not prime, then s( I ) ≤ v p ( I ) ≤ reg( S/I ) , with equality everywhere if S/I is a level ring.Proof.
We set M = S/I , r = reg( S/I ), n = v p ( I ), and I ′ = ( I : p ). To show the inequality n ≤ r we proceed by contradiction. Assume that n > r . The S -modules in the exact sequence0 −→ I ′ /I −→ S/I −→ S/I ′ −→ I ′ /I = 0 (resp. S/I ′ = 0)follows from Proposition 4.2 (resp. I is not prime). That the modules are Cohen–Macaulayfollows observing that I and I ′ are unmixed ideals of dimension 1. Since n is v p ( I ) and r < n ,one has ( I ′ /I ) r = 0 (see the equivalence of Eq. (4.1) in the proof of Proposition 4.2). Hencetaking Hilbert functions in the above exact sequence in degree d = r (resp. for d ≫ S/I ) = H I ( r ) = H I ′ ( r ) ≤ deg( S/I ′ ) (resp. deg( S/I ) = deg( I ′ /I ) + deg( S/I ′ )) . As I ′ /I = 0, deg( I ′ /I ) >
0. Hence deg(
S/I ) > deg( S/I ′ ), a contradiction. Thus n ≤ r . LGEBRAIC INVARIANTS OF GERAMITA IDEALS 15
To show the inequality s( I ) ≤ v p ( I ) we make a change of coefficients. Consider the algebraicclosure K of K . We set S = S ⊗ K K = K [ t , . . . , t s ] and I = IS.
Note that
K ֒ → K is a faithfully flat extension. Apply the functor S ⊗ K ( − ). By base change,it follows that S ֒ → S is a faithfully flat extension. Therefore H I ( d ) = H I ( d ) for d ≥ S/I ) = deg(
S/I ). Furthermore the minimal graded free resolutions and the Hilbert seriesof
S/I and
S/I are identical. Thus
S/I and
S/I have the same regularity, s( I ) = s( I ), and I isCohen–Macaulay of dimension 1. The ideal p = p S is a prime ideal of S because p is generatedby linear forms, and so is p . The ideal I is Geramita. To show this, let I = ∩ mi =1 q i be theminimal primary decomposition of I , where q i is a p i -primary ideal. Since p i S is prime, theideal q i S is a p i S -primary ideal of S , and the minimal primary decomposition of I is I = m \ i =1 q i ! S = m \ i =1 (cid:0) q i S (cid:1) , see [26, Sections 3.H, 5.D and 9.C]. Thus I is a Geramita ideal. Recall that n ≥ f ∈ S n with ( I : f ) = p . Fix f with these two properties. Then f ∈ ( I : p ) \ I and since I ∩ S = I and ( I : p ) S = ( IS : p S ), one has f ∈ ( IS : p S ) \ IS . Therefore,setting p = p S , we obtain v p ( I ) ≤ v p ( I ). Altogether using Proposition 4.9(c), we obtains( I ) = s( I ) ≤ v( I ) ≤ v p ( I ) ≤ v p ( I ) ≤ reg( S/I ) = reg(
S/I ) . If S/I is level then so is
S/I , because the Betti numbers ( b i,j in Definition 2.7) for S/I and
S/I agree [6, 6.10]. Furthermore, since the ring
S/I is level, we have s( I ) = reg( S/I ) by [7,4.13, 4.14] and which gives equality everywhere. (cid:3)
Definition 4.11. [18, 33] Let Z = a P + · · · + a m P m ⊂ P s − be a set of fat points, and supposethat Z ′ = a P + · · · + ( a i − P i + · · · + a m P m for some i = 1 , . . . , m . We call f ∈ S d a separatorof P i of multiplicity a i if f ∈ I ( Z ′ ) \ I ( Z ). The vanishing ideal I ( Z ) of Z is ∩ mi =1 p a i i , where p i is the vanishing ideal of P i . If Z is a set of reduced points (i.e., a = · · · = a m = 1), the degree of P i , denoted deg Z ( P i ), is the least degree of a separator of P i of multiplicity 1. Remark 4.12. If f is a separator of P i of multiplicity a i and p i is the vanishing ideal of P i ,then f ∈ ( I : p i ) \ I . The converse hold if a i = 1. Corollary 4.13. [33, Theorem 3.3]
Let Z = a P + · · · + a m P m ⊂ P s − be a set of fat points,and suppose that Z ′ = a P + · · · + ( a i − P i + · · · + a m P m for some i = 1 , . . . , m . If f is aseparator of P i of multiplicity a i , then deg( f ) ≥ v( I ) ≥ s( I ) .Proof. If f is a separator of P i of multiplicity a i and p i be the vanishing ideal of P i , then f ∈ ( I : p i ) \ I . Hence, by Proposition 4.2 and Theorem 4.10, one has deg( f ) ≥ v( I ) ≥ s( I ). (cid:3) A finite set X = { P , . . . , P m } of reduced points in P s − is Cayley-Bacharach if every hyper-surface of degree less than reg(
S/I ( X )) which contains all but one point of X must contain allthe points of X or equivalently if deg X ( P i ) = reg( S/I ( X )) for all i = 1 , . . . , m [11, Definition 2.7].Since deg X ( P i ) = v p i ( I ), where p i is the vanishing ideal of P i , by Theorem 4.10 one can extendthis notion to Geramita ideals. Definition 4.14.
A Geramita ideal I ⊂ S is called Cayley–Bacharach if v p ( I ) is equal toreg( S/I ) for all p ∈ Ass( I ). As the next result shows Cayley-Bacharach ideals are connected to Reed–Muller type codesand to minimum distance functions.
Corollary 4.15.
A Geramita ideal I ⊂ S is Cayley–Bacharach if and only if reg( δ I ) = v( I ) = reg( S/I ) . Proof.
It follows from Proposition 4.6 and Theorem 4.10. (cid:3)
There are some families of Reed–Muller type codes where the minimum distance and itsindex of regularity are known [22, 31]. In these cases one can determine whether or not thecorresponding sets of points are Cayley–Bacharach.
Corollary 4.16. If K = F q is a finite field and X = P s − , then I ( X ) is Cayley–Bacharach.Proof. It follows from Corollary 4.15 because according to [31] the regularity index of δ I ( X ) isequal to reg( S/I ( X )). (cid:3) Next we give a lemma that allows comparisons between the generalized minimum distancesof ideals related by containment.
Lemma 4.17. If I, I ′ are unmixed graded ideals of the same height and J is a graded ideal suchthat I ′ = ( I : J ), then F d ( I ′ ) ⊂ F d ( I ) and deg( S/I ′ ) − δ I ′ ( d ) ≤ deg( S/I ) − δ I ( d ) . Proof.
Let f ∈ F d ( I ′ ). Then f / ∈ I ′ and ( I ′ : f ) = I ′ , and since we have the following relations I ′ ( ( I ′ : f ) = (( I : J ) : f ) = ( I : ( f J )) = (( I : f ) : J )we deduce that ( I : f ) = I (otherwise the last ideal displayed above would be I ′ ). Note that I ⊂ I ′ , so f / ∈ I . The second statement follows from the inequalitydeg( S/I ′ ) − δ I ′ ( d ) = max { deg( S/ ( I ′ , f )) | f ∈ F d ( I ′ ) }≤ max { deg ( S/ ( I, g )) | g ∈ F d ( I ) } = deg( S/I ) − δ I ( d ) . This inequality is a consequence of the observation that if f ∈ F d ( I ′ ), then ht( I ′ , f ) = ht( I ′ ),and since f ∈ F d ( I ) one also has ht( I, f ) = ht( I ) by Lemma 3.1(a). Thus deg( S/ ( I ′ , f )) ≤ deg( S/ ( I, f )). (cid:3)
One of our main results shows that the function η : N + → Z given by η ( d ) := (deg( S/I ) − H I ( d ) + 1) − δ I ( d )non-negative for Geramita ideals (see Theorem 4.19). Lemma 4.18.
Let I ⊂ S be a Geramita ideal. If F d = ∅ for some d ≥ , then η ( d ) = 0 and η ( d ) ≥ for all d ≥ .Proof. Let p , . . . , p m be the associated primes of I . As p k is generated by linear forms, theinitial ideal of p k , w.r.t the lexicographical order ≺ , is generated by s − p k and in ≺ ( p k ) have the same Hilbert function, deg( S/ p k ) = 1 and H p k ( d ) = 1 for d ≥ F d = ∅ . Then δ I ( d ) = deg( S/I ) and ( I : f ) = I for any f ∈ S d \ I . Hence, byTheorem 2.2, we get ( p ) d ⊂ m [ i =1 p i ! ∩ S d ⊂ I d ⊂ ( p ) d . LGEBRAIC INVARIANTS OF GERAMITA IDEALS 17
Thus I d = ( p ) d , H I ( d ) = H p ( d ) = 1, H I (0) = 1, and η ( d ) = 0. Using Theorem 2.9(ii),one has H I ( d ) = 1 for d ≥
1. Therefore η ( d ) ≥ d ≥ (cid:3) Singleton bound.
We come to one of our main results. The inequality in the followingtheorem is well known when I is the vanishing ideal of a finite set of projective points [30, p. 82].In this case the inequality is called the Singleton bound [34, Corollary 1.1.65].
Theorem 4.19.
Let I ⊂ S be an unmixed graded ideal whose associated primes are generatedby linear forms and such that there exists h ∈ S regular on S/I . If dim(
S/I ) = 1 , then δ I ( d ) ≤ deg( S/I ) − H I ( d ) + 1 for d ≥ or equivalently H I ( d ) − ≤ hyp I ( d ) for d ≥ .Proof. The proof is by induction on deg(
S/I ). If deg(
S/I ) = 1, then I = p is a prime generatedby linear forms, H I ( d ) = 1 for all d ≥ F d ( I ) = ∅ for all d ≥
1. The latter follows since forany prime p , ( p : f ) = p implies f ∈ p . So the result is verified in this case. Let v = v( I ) be thev-number of I . By Proposition 4.2, v = α (( I : p ) /I ) for some p ∈ Ass( I ). Set I ′ = ( I : p ). Theshort exact sequence 0 −→ I ′ /I −→ S/I −→ S/I ′ −→ S/I show that dim ( I ′ /I ) = 1 and depth( I ′ /I ) = 1.Therefore, H I ′ /I ( d ) = 0 for d < α (( I : p ) /I ) = v and H I ′ /I ( d ) > d ≥ α (( I : p ) /I ) = v , andconsequently H I ′ ( d ) = H I ( d ) for d < v and H I ′ ( d ) > H I ( d ) for d ≥ v . The last statement yieldsthat deg( S/I ) > deg( S/I ′ ). This also follows from Lemma 2.10(b).If d < v we deduce from Lemma 4.17, the inductive hypothesis and H I ′ ( d ) = H I ( d ) thatdeg( S/I ) − δ I ( d ) ≥ deg( S/I ′ ) − δ I ′ ( d ) ≥ H I ′ ( d ) − H I ( d ) − , which is the desired inequality. If d ≥ v we know that there exists f ∈ S v such that ( I : f ) = p and thus ( I : h d − v f ) = p . Therefore δ I ( d ) = 1 and since deg( S/I ) ≥ H I ( d ) for any d the desiredinequality follows. (cid:3) The next result is known for complete intersection vanishing ideals over finite fields [15,Lemma 3]. As an application we extend this result to Geramita Gorenstein ideals.
Corollary 4.20.
Let I ⊂ S be a Geramita ideal. If I is Gorenstein and r = reg( S/I ) ≥ ,then δ I ( r − is equal to .Proof. By Proposition 4.6 and Theorem 4.10, r is the regularity index of δ I .Thus δ I ( r − ≥ S/I ) = 1 + H I ( r − K is infinite. Indeed,consider the algebraic closure K of K . We set S = S ⊗ K K and I = IS.
From [32, Lemma 1.1],we have I is Gorenstein, H I ( d ) = H I ( d ) for d ≥ S/I ) = deg(
S/I ). Since K is infinite,there is h ∈ S that is regular on S/I . Then by [2, 3.1.19](b) the quotient ring A = S/ ( I, h )is Gorenstein of dimension 0, by [7, 4.13, 4.14] it has r = reg( S/I ) = reg( A ), and by [2,4.7.11(b)] deg( S/I ) = deg( A ) = P r i =0 H A ( i ). By [2, proof of 4.1.10], F A ( x ) = (1 − x ) F I ( x )hence H I ( n ) = P ni =0 H A ( i ) for any n ≥
0. From here, using that H A ( r ) = 1, since A isGorenstein, we deduce thatdeg( S/I ) = deg(
S/I ) = deg( A ) = r X i =0 H A ( i ) = 1 + r − X i =0 H A ( i ) = 1 + H I ( r −
1) = 1 + H I ( r − . Finally, making d = r − δ I ( r − ≤
2. Thus equality holds. (cid:3)
Note that the situation is quite different from the conclusion of Theorem 4.19 if dim(
S/I ) ≥ Proposition 4.21.
Let I ⊂ S be an unmixed graded ideal. If dim( S/I ) ≥ , then δ I ( d ) > deg( S/I ) − H I ( d ) + 1 for some d ≥ . Proof.
Note that m = ( t , . . . , t s ) is not an associated prime of I , that is, depth( S/I ) ≥ F d = ∅ for some d ≥
2. As H I (0) = 1 and δ I ( d ) is equal to deg( S/I ), byTheorem 2.9(i), one has H I ( d ) > F d = ∅ for d ≥ d ≥ f d ∈ F d such that δ I ( d ) = deg( S/I ) − deg( S/ ( I, f d )) . As H I is strictly increasing by Theorem 2.9(i), using Lemma 3.1(b), we getdeg( S/ ( I, f d )) < deg( S/I ) < H I ( d ) − d ≫
0. Thus the required inequality holds for d ≫ (cid:3) Reed-Muller type codes
In this section we give refined information on the minimum distance function for the Reed–Muller codes defined in the Introduction. The key insight is that, in the case of the projectiveReed–Muller codes, this minimum distance function can be realized as a generalized minimumdistance function for a finite set of points in projective space, often called evaluation points inthe algebraic coding context.
Theorem 5.1. [14, Theorem 4.5]
Let X be a finite set of points in a projective space P s − overa field K and let I ( X ) be its vanishing ideal. If d ≥ and ≤ r ≤ H X ( d ) , then δ r ( C X ( d )) = δ I ( X ) ( d, r ) . By Theorem 3.8 (a) and [37, Theorem 1, Corollary 1], the entries of each row of the weightmatrix ( δ X ( d, r )) form an increasing sequence until they stabilize. We show in Theorem 5.3 belowthat the entries of each column of the weight matrix ( δ X ( d, r )) form a decreasing sequence.Before we can prove this result we need an additional lemma. Recall that the support χ ( β )of a vector β ∈ K m is χ ( Kβ ), that is, χ ( β ) is the set of non-zero entries of β . Lemma 5.2.
Let D be a subcode of C of dimension r ≥ . If β , . . . , β r is a K -basis for D with β i = ( β i, , . . . , β i,m ) for i = 1 , . . . , r , then χ ( D ) = ∪ ri =1 χ ( β i ) and the number of elements of χ ( D ) is the number of non-zero columns of the matrix: β , · · · β ,i · · · β ,m β , · · · β ,i · · · β ,m ... · · · ... · · · ... β r, · · · β r,i · · · β r,m . Theorem 5.3.
Let X be a finite set of points in P s − , let I = I ( X ) be its vanishing ideal, andlet ≤ r ≤ | X | be a fixed integer. Then there is an integer d ≥ such that δ I (1 , r ) > δ I (2 , r ) > · · · > δ I ( d , r ) = δ I ( d, r ) = r for d ≥ d . Proof.
Let [ P ] , . . . , [ P m ] be the points of X . By Theorem 5.1 there exists a linear subcode D of C X ( d ) of dimension r such that δ I ( d, r ) = δ X ( d, r ) = | χ ( D ) | . Pick a K -basis β , . . . , β r of D .Each β i can be written as β i = ( β i, , . . . , β i,k , . . . , β i,m ) = ( f i ( P ) , . . . , f i ( P k ) , . . . , f i ( P m )) LGEBRAIC INVARIANTS OF GERAMITA IDEALS 19 for some f i ∈ S d . Consider the matrix B whose rows are β , . . . , β m : B = f ( P ) · · · f ( P k ) · · · f ( P m ) f ( P ) · · · f ( P k ) · · · f ( P m )... · · · ... · · · ... f r ( P ) · · · f r ( P k ) · · · f r ( P m ) . As B has rank r , by permuting columns and applying elementary row operations, the matrix B can be brought to the form: B ′ = g ( Q ) g ( Q r +1 ) · · · g ( Q m ) g ( Q ) g ( Q r +1 ) · · · g ( Q m ) . . . ... g r ( Q r ) g r ( Q r +1 ) · · · g r ( Q m ) , where g , . . . , g r are linearly independent polynomials over the field K modulo I of degree d , Q , . . . , Q m are a permutation of P , . . . , P m , the first r columns of B ′ form a diagonal matrixsuch that g i ( Q i ) = 0 for i = 1 , . . . , r , and the ideals ( f , . . . , f r ) and ( g , . . . , g r ) are equal. Let D ′ be the linear space generated by the rows of B ′ . The operations applied to B did not affectthe size of the support of D (Lemma 5.2), that is, | χ ( D ) | = | χ ( D ′ ) | .Note that δ r ( C X ( d )) depends only on X , that is, δ r ( C X ( d )) is independent of how we or-der the points in X (cf. Theorem 5.1). Let ev ′ d : S d → K m be the evaluation map, f ( f ( Q ) , . . . , f ( Q m )), relative to the points [ Q ] , . . . , [ Q m ]. By Theorem 3.8, δ X ( d, r ) ≥ r .First we assume that δ X ( d, r ) = r for some d ≥ r ≥
1. Then the i -th column of B ′ is zero for i > r . For each 1 ≤ i ≤ r pick h i ∈ S such that h i ( Q i ) = 0. The polynomials h g , . . . , h r g r are linearly independent modulo I because ( h i g i )( Q j ) is not 0 if i = j and is 0 if i = j . The image of Kh g ⊕ · · · ⊕ Kh r g r , under the map ev ′ d +1 , is a subcode D ′′ of C X ( d + 1)of dimension r and | χ ( D ′′ ) | = r . Thus δ X ( d + 1 , r ) ≤ r , and consequently δ X ( d + 1 , r ) = r .Next we assume that δ X ( d, r ) > r . Then B ′ has a nonzero column ( g ( Q k ) , . . . , g r ( Q k )) ⊤ forsome k > r . It suffices to show that δ X ( d, r ) > δ X ( d + 1 , r ). According to [24, Lemma 2.14(ii)]for each 1 ≤ i ≤ r there is h i in S such that h i ( Q i ) = 0 and h i ( Q k ) = 0. Let B ′′ be the matrix: B ′′ = h g ( Q ) h g ( Q r +1 ) · · · h g ( Q m ) h g ( Q ) h g ( Q r +1 ) · · · h g ( Q m ) . . . ... h r g r ( Q r ) h r g r ( Q r +1 ) · · · h r g r ( Q m ) . The image of Kh g ⊕ · · · ⊕ Kh r g r , under the map ev ′ d +1 , is a subcode V of C X ( d + 1) ofdimension r because the rank of B ′′ is r , and since the k -column of B ′′ is zero, we get δ X ( d, r ) = | χ ( D ) | = | χ ( D ′ ) | > | χ ( V ) | ≥ δ X ( d + 1 , r ) . Thus δ X ( d, r ) > δ X ( d + 1 , r ). (cid:3) Corollary 5.4.
Let X be a finite set of points in P s − and let I = I ( X ) be its vanishing ideal.If I is a complete intersection, then δ I ( d ) ≥ reg( S/I ) − d + 1 for ≤ d < reg( S/I ) .Proof. If r denotes the regularity of S/I , by Theorem 4.10, one has v(I) = r . Thus δ I ( r − ≥ r = 1. (cid:3) Corollary 5.5. [13, Theorem 12] If X is a set parameterized by monomials lying on a projectivetorus and ≤ r ≤ | X | be a fixed integer, then there is an integer d ≥ such that δ r ( C X (1)) > δ r ( C X (2)) > · · · > δ r ( C X ( d )) = δ r ( C X ( d )) = r for d ≥ d . Proof.
It follows at once from Theorems 5.1 and 5.3. (cid:3)
Corollary 5.6.
Let X be a finite set of points of P s − and let δ X ( d ) be the minimum distanceof C X ( d ) . Then δ X ( d ) = 1 if and only if d ≥ v( I ) .Proof. It follows from Proposition 4.6, and Theorems 5.1 and 5.3. (cid:3)
Corollary 5.7. If X is a finite set of P s − over a field K , then δ X ( d, H X ( d )) = | X | for d ≥ .Proof. It follows at once from Proposition 3.14 and Theorem 5.1. (cid:3) Complete intersections
In this section we examine minimum distance functions of complete intersection ideals.
Definition 6.1.
An ideal I ⊂ S is called a complete intersection if there exist g , . . . , g r in S such that I = ( g , . . . , g r ), where r = ht( I ) is the height of I .There are a number of interesting open problems regarding the minimum distance of completeintersection functions. We discuss one such problem in Conjecture 6.2 below and relate thisproblem to [8, Conjecture CB12]) in the second part of this section. Conjecture 6.2. [25]
Let I ⊂ S := K [ t , . . . , t s ] be a complete intersection graded ideal ofdimension generated by forms f , . . . , f c , c = s − , with d i = deg( f i ) and ≤ d i ≤ d i +1 for i ≥ . If the associated primes of I are generated by linear forms, then δ I ( d ) ≥ ( d k +1 − ℓ ) d k +2 · · · d c if ≤ d ≤ c X i =1 ( d i − − , where ≤ k ≤ c − and ℓ are integers such that d = P ki =1 ( d i −
1) + ℓ and ≤ ℓ ≤ d k +1 − . This conjecture holds if the initial ideal of I with respect to some monomial order is a completeintersection [25, Theorem 3.14]. Our results show that for complete intersections v( I ) = reg( S/I )(Theorem 4.10) and δ I ( d ) ≥ reg( S/I ) − d + 1 for 1 ≤ d < reg( S/I ) if I is a vanishing ideal(Corollary 5.4). Thus the conjecture is best possible for vanishing ideals in the sense that itcovers all cases where δ I ( d ) > S/I is equal to c P i =1 ( d i − Conjecture 6.3.
Let X be a finite set of reduced points in P s − and suppose that I = I ( X ) is acomplete intersection generated by f , . . . , f c , c = s − , with d i = deg( f i ) for i = 1 , . . . , c , and ≤ d i ≤ d i +1 for all i . Then (a) [33, Conjecture 4.9] δ I (1) ≥ ( d − d · · · d c . (b) If f , . . . , f c are quadratic forms, then δ I ( d ) ≥ c − d for ≤ d ≤ c or equivalently hyp I ( d ) ≤ c − c − d for ≤ d ≤ c . We prove part (a) of this conjecture, in a more general setting, when I is equigenerated. LGEBRAIC INVARIANTS OF GERAMITA IDEALS 21
Proposition 6.4.
Let I ⊂ S be an unmixed graded ideal of height c , minimally generated byforms of degree e ≥ , whose associated primes are generated by linear forms. Then hyp I (1) ≤ e c − and δ I (1) ≥ deg( S/I ) − e c − . Furthermore δ I (1) ≥ e c − e c − if I is a complete intersection.Proof. Since the associated primes of I are generated by linear forms and e ≥
2, one has F ( I ) = ∅ . Take any linear form h = t k − P j = i λ j t j in F ( I ), λ j ∈ K . For simplicity of notationassume k = 1. It suffices to show that deg( S/ ( I, h )) ≤ e c − . Let { f , . . . , f n } be a minimal setof generators of I consisting of homogeneous polynomials with deg( f i ) = e for all i . Setting f ′ i = f i ( P j =1 λ j t j , t , . . . , t s ) for i = 1 , . . . , n , S ′ = K [ t , . . . , t s ], and I ′ = ( f ′ , . . . , f ′ n ), there isan isomorphism ϕ of graded K -algebras S/ ( I, h ) ϕ −→ S ′ /I ′ , t λ t + · · · + λ s t s , t i t i , i = 2 , . . . , s. Note that ϕ ( f + ( I, h )) = f ( λ t + · · · + λ s t s , t , . . . , t s ) + I ′ for f in S and that ϕ has degree0, that is, ϕ is degree preserving. Hence S/ ( I, h ) and deg( S ′ /I ′ ) have the same degree and thesame dimension. Since ht( I, h ) = ht( I ), we get ht( I ′ ) = ht( I ) −
1, that is, ht( I ′ ) = c −
1. Bydefinition f ′ i is either 0 or has degree e , that is, I ′ is generated by forms of degree e . As K isinfinite, there exists a minimal set of generators of I ′ , { g , . . . , g t } , such that deg( g i ) = e for all i and g , . . . , g c − form a regular sequence (see Lemma 2.11). From the exact sequence0 −→ I ′ / ( g , . . . , g c − ) −→ S ′ / ( g , . . . , g c − ) −→ S ′ /I ′ −→ , we get e c − = deg( S/ ( g , . . . , g c − )) ≥ deg( S ′ /I ′ ) = deg( S/ ( I, h )). This proves that hyp I (1)is less than or equal to e c − . Hence δ I (1) ≥ deg( S/I ) − e c − . Therefore, if I is a completeintersection, deg( S/I ) = e c and we obtain the inequality δ I (1) ≥ e c − e c − . (cid:3) As a consequence, we recover the fact that Conjecture 6.3(a) holds for P [33, Theorem 4.10]. Corollary 6.5.
Let I ⊂ S be a graded ideal of height , minimally generated by two forms f , f of degrees e , e , with ≤ e ≤ e , whose associated primes are generated by linear forms. Then hyp I (1) ≤ e and δ I (1) ≥ e e − e .Proof. It follows adapting the proof of Proposition 6.4. (cid:3)
Corollary 6.6.
Let I ⊂ S be an unmixed graded ideal minimally generated by forms of degree e ≥ whose associated primes are generated by linear forms. If ≤ r ≤ ht( I ) , then hyp I (1 , r ) ≤ e ht( I ) − r ,δ I (1 , r ) ≥ deg( S/I ) − e ht( I ) − r , and δ I (1 , r ) ≥ e ht( I ) − e ht( I ) − r if I is a complete intersection.Proof. This follows by adapting the proof of Proposition 6.4 and observing the following. If f , . . . , f r are linearly independent linear forms and t ≻ · · · ≻ t s is the lexicographical order,we can find linear forms h , . . . , h r such that in ≺ ( h ) ≻ · · · ≻ in ≺ ( h r ) and ( f , . . . , f r ) is equalto ( h , . . . , h r ). (cid:3) Cayley-Bacharach Conjectures.
In the following we explore the connections between a mod-ified form of Conjecture 6.2 and a conjecture of Eisenbud-Green-Harris [8, Conjecture CB12].
Conjecture 6.7 (Strong form of [8, Conjecture CB12] ) . Let Γ be any subscheme of a zero-dimensional complete intersection of hypersurfaces of degrees d ≤ · · · ≤ d c in a projective space P c . If Γ fails to impose independent conditions on hypersurfaces of degree m , then deg(Γ) ≥ ( e + 1) d k +2 d k +3 · · · d c where e and k are defined by the relations c X i = k +2 ( d i − ≤ m + 1 < c X i = k +1 ( d i − and e = m + 1 − c X i = k +2 ( d i − . Proposition 6.8.
Conjectures 6.2 and 6.7 are equivalent for radical complete intersections.Proof.
We first prove that Conjecture 6.7 for m = P ci = k +1 ( d i − − ℓ − e = d k +1 − ℓ − I be a radical complete intersection ideal minimally generated byforms of degrees d ≤ · · · ≤ d c . Let H be any hypersurface defined by a form F of degree d . Let X be the scheme defined by I ( X ) = ( I, F ) and let Γ be the residual scheme defined by I (Γ) = I : F . By the Cayley-Bacharach Theorem [8, CB7], Γ must fail to impose independentconditions on hypersurfaces of degree P ci =1 ( d i − − d − m . Now Conjecture 6.7 impliesdeg( S/I : F ) = deg(Γ) ≥ ed k +2 d k +3 · · · d c , which in view of Theorem 3.5 gives δ I ( d ) ≥ ( e + 1) d k +2 d k +3 · · · d c = ( d k +1 − ℓ ) d k +2 d k +3 · · · d c . For the converse, we prove that Conjecture 6.2 with d = P ci =1 ( d i − − m − ℓ = d k +1 − e − m . Assuming that Γ spansa projective space P c , take a radical complete intersection ideal I contained in I Γ , and let X be the scheme defined by I ( X ) = I : I (Γ). By [8, CB7], X lies on a hypersurface of degree P ci =1 ( d i − − m − d . Then Conjecture 6.2 and Theorem 3.5 givedeg(Γ) = deg( S/I : F ) ≥ ( d k +1 − ℓ ) d k +2 d k +3 · · · d c = ( e + 1) d k +2 d k +3 · · · d c . (cid:3) Conjecture 6.7 has been recently proven in [20, Theorem 5.1] for k = 1 under additionalassumptions on the Picard group of the complete intersection. We now consider the case when d = · · · = d c = 2. In this case Conjecture 6.2 specializes to Conjecture 6.3(b) and Conjecture6.7 is related to [8, Conjecture CB10]. Proposition 6.9.
The following statements are equivalent: (1) [Conjecture 6.3(b)] Let I be a complete intersection generated by c quadratic forms. Then δ I ( d ) ≥ c − d for ≤ d ≤ c or equivalently hyp I ( d ) ≤ c − c − d for ≤ d ≤ c . (2) [8, Conjecture CB10] If X is an ideal-theoretic complete intersection of c = s − quadricsin P s − and f ∈ S := K [ t , . . . , t s ] is a homogeneous polynomial of degree d such that deg( S/ ( I ( X ) , f )) > c − c − d , then f ∈ I X .Proof. Let I = I ( X ) be a complete intersection ideal of c quadratic homogeneous polynomials.Then deg( S/I ) = 2 c and (2) is equivalent to the statement for any f ∈ F d , deg( S/ ( I ( X ) , f )) ≤ c − c − d . Using Definition 1.1, this is in turn equivalent to δ I ( d ) ≥ c − d , for d = 1 , . . . , c , whichis precisely the statement of (1). (cid:3) As an application of our earlier results we recover the following cases of Conjecture 6.3(b)under the more general hypothesis that I ( X ) is a not necessarily a radical complete intersection. LGEBRAIC INVARIANTS OF GERAMITA IDEALS 23
Corollary 6.10. [8] If I is a complete intersection ideal generated by c quadratic forms, then δ I ( d ) ≥ c − d , for d = 1 , c − , and c Proof.
Case d = 1 follows by taking e = 2 in Proposition 6.4. Case d = c − d = c recall that By Proposition 4.6 and Theorem 4.10,reg( S/I ) is the regularity index of δ I . Thus Case d = c follows since reg( S/I ) = c . (cid:3) If X ⊂ P c is a reduced set of points such that I ( X ) is a complete intersection ideal and thepoints of X are in linearly general position, i.e., any c + 1 points of X span P c , then [1, Theorem1] adds more cases when [8, Conjecture CB10] holds. If we assume that the quadrics that cutout X are generic, then the assumptions of that result are satisfied, and we can conclude byProposition 6.9 that if 1 ≤ d ≤ c −
1, then δ I ( d ) ≥ c ( c − − d ) + 2. Suppose d := c − k , for some1 ≤ k ≤ c −
1. Then this inequality becomes δ I ( c − k ) ≥ c ( k −
1) + 2. We observe that thisinequality is sufficiently strong to establish [Conjecture 6.3(b)] asymptotically for c sufficientlylarge: • k = 2. If c ≥
2, then δ I ( c − ≥ c + 2 ≥ . • k = 3. If c ≥
3, then δ I ( c − ≥ c + 2 ≥ . • k = 4. If c ≥
5, then δ I ( c − ≥ c + 2 ≥ . • k ≥
5. If c ≥ (2 k − / ( k − δ I ( c − k ) ≥ k . Acknowledgements.
We thank the Banff International Research Station (BIRS) where this project was started dur-ing under the auspices of their focused research groups program. Computations with
Macaulay2 [17] were crucial to verifying many of the examples given in this paper. The authors acknowl-edge the support of the following funding agencies: Cooper was partially funded by NSERCgrant RGPIN-2018-05004, Seceleanu was partially supported by NSF grant DMS–1601024 andEPSCoR grant OIA–1557417.
Appendix A. Procedures for
Macaulay Procedure A.1.
Computing the footprint matrix with
Macaulay I to obtain the entries of thematrix (fp I ( d, r )) and is reasonably fast. S=QQ[t1,t2,t3], I=ideal(t1^3,t2*t3)M=coker gens gb Iregularity M, degree M, init=ideal(leadTerm gens gb I)er=(x)-> if not quotient(init,x)==init then degree ideal(init,x) else 0fpr=(d,r)->degree M - max apply(apply(apply(subsets(flatten entries basis(d,M),r),toSequence),ideal),er)hilbertFunction(1,M),fpr(1,1),fpr(1,2),fpr(1,3)--gives the first row of the footprint matrix
Procedure A.2.
Computing the GMD function with
Macaulay q=3,S=ZZ/3[t1,t2,t3,t4,t5,t6],I=ideal(t1*t6-t3*t4,t2*t6-t3*t5)G=gb I, M=coker gens gb I regularity M, degree M, init=ideal(leadTerm gens gb I)genmd=(d,r)->degree M-max apply(apply(subsets(apply(apply(apply(toList (set(0..q-1))^**(hilbertFunction(d,M))-(set{0})^**(hilbertFunction(d,M)),toList),x->basis(d,M)*vector x),z->ideal(flatten entries z)),r),ideal),x-> if
Procedure A.3.
Computing the minimum socle degree and the v-number of an ideal I withMacaulay2 [17]. This procedure corresponds to Example 4.3. S=QQ[t1,t2,t3,t4]p1=ideal(t2,t3,t4),p2=ideal(t1,t3,t4),p3=ideal(t1,t2,t4),p4=ideal(t1,t2,t3)I=intersect(ideal(t2^10,t3^9,t4^4,t2*t3*t4^3),ideal(t1^4,t3^4,t4^3,t1*t3*t4^2),ideal(t1^4,t2^5,t4^3),ideal(t1^3,t2^5,t3^10))h=ideal(t1+t2+t3+t4)--regular element on S/IJ=quotient(I+h,m), regularity coker gens gb Isoc=J/(I+h), degrees mingens socJ1=quotient(I,p1), soc1=J1/I, degrees mingens soc1
Procedure A.4.
Computing the v-number of a vanishing ideal I ( X ), the regularity index of δ X , and the minimum distance δ X ( d ) of the Reed–Muller-type code C X ( d ) with Macaulay2 [17].This procedure corresponds to Example 4.5. q=3, G=ZZ/q, S=G[t3,t2,t1,MonomialOrder=>Lex]p1=ideal(t2,t1-t3),p2=ideal(t2,t3),p3=ideal(t2,2*t1-t3)p4=ideal(t1-t2,t3),p5=ideal(t1-t3,t2-t3), p6=ideal(2*t1-t3,2*t2-t3)p7=ideal(t1,t2),p8=ideal(t1,t3),p9=ideal(t1,t2-t3),p10=ideal(t1,2*t2-t3)I=intersect(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10)M=coker gens gb I, regularity M, degree Minit=ideal(leadTerm gens gb I)genmd=(d,r)->degree M-max apply(apply(subsets(apply(apply(apply(toList (set(0..q-1))^**(hilbertFunction(d,M))-(set{0})^**(hilbertFunction(d,M)),toList),x->basis(d,M)*vector x),z->ideal(flatten entries z)),r),ideal),x-> if LGEBRAIC INVARIANTS OF GERAMITA IDEALS 25
References [1] E. Ballico and C. Fontanari, The Horace method for error-correcting codes,
AAECC (2006), 135–139.[2] W. Bruns and J. Herzog, Cohen-Macaulay Rings , Revised Edition, Cambridge University Press, 1997.[3] C. Carvalho, On the second Hamming weight of some Reed-Muller type codes, Finite Fields Appl. (2013),88–94.[4] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms , Springer-Verlag, 1992.[5] I. M. Duursma, C. Renter´ıa and H. Tapia-Recillas, Reed-Muller codes on complete intersections, Appl.Algebra Engrg. Comm. Comput. (2001), no. 6, 455–462.[6] D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry , Graduate Texts in Mathematics , Springer-Verlag, 1995.[7] D. Eisenbud,
The geometry of syzygies: A second course in commutative algebra and algebraic geometry ,Graduate Texts in Mathematics , Springer-Verlag, New York, 2005.[8] D. Eisenbud, M. Green and J. Harris, Cayley–Bacharach theorems and conjectures, Bull. Amer. Math. Soc.(N.S.) (1996), no. 3, 295–324.[9] B. Engheta, On the projective dimension and the unmixed part of three cubics, J. Algebra (2007), no.2, 715–734.[10] R. Fr¨oberg and D. Laksov, Compressed Algebras , Lecture Notes in Mathematics (1984), Springer-Verlag,pp. 121–151.[11] A. V. Geramita, M. Kreuzer and L. Robbiano, Cayley–Bacharach schemes and their canonical modules,
Trans. Amer. Math. Soc. (1993), no. 1, 163–189.[12] L. Gold, J. Little and H. Schenck, Cayley-Bacharach and evaluation codes on complete intersections, J. PureAppl. Algebra (2005), no. 1, 91–99.[13] M. Gonz´alez-Sarabia, E. Camps, E. Sarmiento and R. H. Villarreal, The second generalized Hamming weightof some evaluation codes arising from a projective torus, Finite Fields Appl. (2018), 370–394.[14] M. Gonz´alez-Sarabia, J. Mart´ınez-Bernal, R. H. Villarreal and C. E. Vivares, Generalized minimum distancefunctions, J. Algebraic Combin., to appear.[15] M. Gonz´alez-Sarabia and C. Renter´ıa, The dual code of some Reed–Muller type codes, Appl. Algebra Engrg.Comm. Comput. (2004), 329–333.[16] M. Gonz´alez-Sarabia, C. Renter´ıa and H. Tapia-Recillas, Reed-Muller-type codes over the Segre variety,Finite Fields Appl. (2002), no. 4, 511–518.[17] D. Grayson and M. Stillman, Macaulay a software system for research in algebraic geometry , available at .[18] E. Guardo, L. Marino and A. Van Tuyl, Separators of fat points in P n , J. Algebra (2010), no. 7,1492–1512.[19] T. Helleseth, T. Kløve and J. Mykkelveit, The weight distribution of irreducible cyclic codes with blocklengths n (( q l − /N ), Discrete Math. (1977), 179–211.[20] J. Hotchkiss and B. Ullery, The gonality of complete intersection curves, Preprint, 2018, arXiv:1706.08169.[21] T. Kløve, The weight distribution of linear codes over GF ( q l ) having generator matrix over GF ( q ), DiscreteMath. (1978), no. 2, 159–168.[22] H. H. L´opez, C. Renter´ıa and R. H. Villarreal, Affine cartesian codes, Des. Codes Cryptogr. (2014), no.1, 5–19.[23] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, 1977.[24] J. Mart´ınez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of graded ideals and Reed-Muller-type codes, J. Pure Appl. Algebra (2017), 251–275.[25] J. Mart´ınez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of complete intersections,J. Algebra Appl. (2018), no. 11, 1850204 (22 pages).[26] H. Matsumura, Commutative Algebra , Benjamin-Cummings, Reading, MA, 1980.[27] H. Matsumura,
Commutative Ring Theory , Cambridge Studies in Advanced Mathematics , CambridgeUniversity Press, 1986.[28] L. N´u˜nez-Betancourt, Y. Pitones and R. H. Villarreal, Footprint and minimum distance functions, Commun.Korean Math. Soc. (2018), no. 1, 85–101.[29] L. O’Carroll, F. Planas-Vilanova and R. H. Villarreal, Degree and algebraic properties of lattice and matrixideals, SIAM J. Discrete Math. (2014), no. 1, 394–427.[30] C. Renter´ıa, A. Simis and R. H. Villarreal, Algebraic methods for parameterized codes and invariants ofvanishing ideals over finite fields, Finite Fields Appl. (2011), no. 1, 81–104.[31] A. Sørensen, Projective Reed-Muller codes, IEEE Trans. Inform. Theory (1991), no. 6, 1567–1576. [32] R. Stanley, Hilbert functions of graded algebras, Adv. Math. (1978), 57–83.[33] S. Tohˇaneanu and A. Van Tuyl, Bounding invariants of fat points using a coding theory construction, J. PureAppl. Algebra (2013), no. 2, 269–279.[34] M. Tsfasman, S. Vladut and D. Nogin, Algebraic geometric codes : basic notions , Mathematical Surveys andMonographs , American Mathematical Society, Providence, RI, 2007.[35] W. V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry , Springer-Verlag, 1998.[36] R. H. Villarreal,
Monomial Algebras, Second Edition , Monographs and Research Notes in Mathematics,Chapman and Hall/CRC, 2015.[37] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory (1991), no. 5,1412–1418. Department of Mathematics, 420 Machray Hall, 186 Dysart Road University of Manitoba, Win-nipeg, MB R3T 2N2 Canada.
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