Minimum distance functions of complete intersections

0.26 MB·

by  Yuriko Pitones

MINIMUM DISTANCE FUNCTIONS OF COMPLETE INTERSECTIONS
JOSE´ MART´INEZ-BERNAL,YURIKOPITONES,AND RAFAEL H.VILLARREAL
6
Abstract. Westudytheminimumdistancefunction ofacompleteintersectiongradedidealin
1
a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial
0
ideal is a complete intersection, we use the footprint function to give a sharp lower bound for
2
theminimum distance function. Then we show some applications to coding theory.
t
c
O

2
1. Introduction
1

C] LetS = K[t1,...,ts] = ⊕∞d=0Sd beapolynomial ringover a fieldK withthestandardgrading
and let I 6= (0) be a graded ideal of S. The degree or multiplicity of S/I is denoted by deg(S/I).
A
Fix a graded monomial order ≺ on S and let in (I) be the initial ideal of I.
≺
.
h Thefootprint of S/I or Gr¨obner ´escalier of I, denoted ∆ (I), is the set of all monomials of S
≺
t
a not in the ideal in≺(I) [28, p. 13, p. 133]. This notion occurs in other branches of mathematics
m under different names; see [20, p. 6] for a list of alternative names.
[
Given an integer d ≥ 1, let M be the set of all zero-divisors of S/in (I) of degree d that
≺,d ≺

3 are in ∆ (I), and let F be the set of all zero-divisors of S/I that are not zero and are a
≺ ≺,d
v K-linear combination of monomials in ∆ (I) of degree d.
≺
4
0 The footprint function of I, denoted fpI, is the function fpI: N+ → Z given by
6
deg(S/I)−max{deg(S/(in (I),ta))|ta ∈ M } if M 6= ∅,
7 fp (d) := ≺ ≺,d ≺,d
0 I deg(S/I) if M≺,d = ∅,
(cid:26)
.
1 and the minimum distance function of I, denoted δ , is the function δ : N → Z given by
I I +
0
6 δ (d) := deg(S/I)−max{deg(S/(I,f))|f ∈F≺,d} if F≺,d 6= ∅,
1 I deg(S/I) if F = ∅.
≺,d
: (cid:26)
v
These two functions were introduced and studied in [24]. Notice that δ is independent of the
i I
X
monomial order ≺ (see Lemma 3.9). To compute δ is a difficult problem but to compute fp is
I I
r much easier.
a
We come to the main result of this paper which gives an explicit lower bound for δ and a
I
formula for fp for a family of complete intersection graded ideals:
I
Theorem 3.14 If the initial ideal in (I) of I is a complete intersection of height s−1 generated
≺
by tα2,...,tαs, with di = deg(tαi) and 1≤ di ≤ di+1 for i ≥ 2, then
s
(d −ℓ)d ···d if d≤ (d −1)−1,
k+2 k+3 s i
δI(d) ≥ fpI(d) =  i=s2
 1 if d≥ P(di −1),
i=2
where 0≤ k ≤ s−2 and ℓ are integers such that d= k+1(d −P1)+ℓ and 1 ≤ ℓ ≤ d −1.
 i=2 i k+2
2010 Mathematics Subject Classification. Primary 13P25; SePcondary 94B60, 11T71.
The first and third author were supported by SNI.The second authorwas supported by CONACyT.
1
2 JOSE´ MART´INEZ-BERNAL,YURIKOPITONES,ANDRAFAELH.VILLARREAL
An important case of this theorem, from the viewpoint of applications, is when I is the
vanishing ideal of a finite set of projective points over a finite field (see the discussion below
abouttheconnection of fp andδ withcodingtheory). If I is acomplete intersection monomial
I I
ideal of dimension 1, then δ (d) = fp (d) for d ≥ 1 (see Proposition 3.11), but this case is only
I I
of theoretical interest because, by Proposition 2.16, a monomial ideal is a vanishing ideal only
in particular cases.
Let I ⊂ S be a graded ideal such that L = in (I) is a complete intersection of dimension 1.
≺
We give a formula for the degree of S/(L,ta) when ta is in M , that is, ta is not in L and is
≺,d
a zero-divisor of S/L. By an easy classification of the complete intersection property of L (see
Lemma 3.1) there are basically two cases to consider. One of them is Lemma 3.4, and the other
is the following:
Lemma 3.3 If L = in (I) is generated by td2,...,tds and ta = ta1···tas is in M , then
≺ 2 s 1 s ≺,d
deg(S/(L,ta)) = d ···d −(d −a )···(d −a ).
2 s 2 2 s s
To show our main result we use the formula for the degree of the ring S/(L,ta), and then
use Proposition 3.13 to bound the degrees uniformly. The proof of the main result takes place
in an abstract algebraic setting with no reference to vanishing ideals or finite fields.
The formulas for the degree are useful in the following setting. If I = I(X) is the vanishing
idealofafinitesetXofprojectivepoints,andin (I)isgeneratedbytd2,...,tds,thenLemma3.3
≺ 2 s
can be used to give upper bounds for the number of zeros in X of homogeneous polynomials of
S. In fact, if f ∈ F and in (f) = ta1···tas, then in (f) is in M , and by Corollary 3.8
≺,d ≺ 1 s ≺ ≺,d
one has:
|V (f)|≤ d ···d −(d −a )···(d −a ),
X 2 s 2 2 s s
where V (f) is the set of zeros or variety of f in X. This upper bound depends on the exponent
X
of theleading term of f. Amore complex upperboundis obtained whenthe initial ideal of I(X)
is as in Lemma 3.1(ii). In this case one uses the formula for the degree given in Lemma 3.4.
Theinterest in studyingfp and δ comes from algebraic codingtheory. Indeed, if I = I(X) is
I I
the vanishing ideal of a finite subset X of a projective space Ps−1 over a finite field K = F , then
q
the minimum distance δ (d) of the corresponding projective Reed-Muller-type code is equal to
X
δ (d), and fp (d) is a lower bound for δ (d) for d≥ 1 (see Theorem 2.13 and Lemma 3.10).
I(X) I(X) X
Therefore, one has the formula:
(2.2) δ (d) = deg(S/I(X))−max{|V (f)|: f 6≡ 0;f ∈ S },
I(X) X d
where f 6≡ 0 means that f is not the zero function on X. Our abstract study of the minimum
distance and footprint functions provides fresh techniques to study δ (d).
X
It is well-known that the degree of S/I(X) is equal to |X| [18, Lecture 13]. Hence, using
Eq.(2.2) and ourmain result, we get thefollowing uniform upperboundfor the numberof zeros
of all polynomials f ∈ S that do not vanish at all points of X.
d
Corollary 4.2 If the initial ideal in (I(X)) is a complete intersection generated by tα2,...,tαs,
≺
with di = deg(tαi) and 1 ≤ di ≤ di+1 for i ≥ 2, then
(4.1) |V (f)| ≤ |X|−(d −ℓ)d ···d ,
X k+2 k+3 s
for any f ∈ S that does not vanish at all point of X, where 0 ≤ k ≤ s−2 and ℓ are integers
d
such that d= k+1(d −1)+ℓ and 1 ≤ ℓ ≤ d −1.
i=2 i k+2
This result gives a tool for finding good uniform upper bounds for the number of zeros in X
P
of polynomials over finite fields. This is a problem of fundamental interest in algebraic coding
COMPLETE INTERSECTIONS 3
theory [31] and algebraic geometry [30]. We leave as an open question whether this uniform
bound is optimal, that is, whether the equality is attained for some polynomial f.
Tohaˇneanu and Van Tuyl conjectured [33, Conjecture 4.9] that if the vanishing ideal I(X) is
a complete intersection generated by polynomials of degrees d ,...,d and d ≤ d for all i,
2 s i i+1
then δ (1) ≥(d −1)d ...d . By Corollary 4.2 this conjecture is true if in (I(X)) is a complete
X 2 3 s ≺
intersection. We leave as another open question whether Corollary 4.2 is true if we only assume
that I(X) is a complete intersection (cf. Proposition 3.12).
ToillustratetheuseofCorollary 4.2inaconcretesituation, considerthelexicographical order
on S with t ≺ ··· ≺ t and a projective torus over a finite field F with q 6= 2:
1 s q
T = {[(x ,...,x )] ∈ Ps−1|x ∈ F∗ for i= 1,...,s},
1 s i q
where F∗ = F \{0}. As I(T) is generated by the Gro¨bner basis {tq−1−tq−1}s , its initial ideal
q q i 1 i=2
is a complete intersection generated by tq−1,...,tq−1. Therefore, noticing that deg(S/I(T)) is
2 s
equal to (q − 1)s−1 and setting d = q − 1 for i = 2,...,s in Eq. (4.1), we obtain that any
i
homogeneous polynomial f of degree d, not vanishing at all points of T, has at most
(q−1)s−1−(q−1)s−(k+2)(q−1−ℓ)
zeros in T if d ≤ (q − 2)(s − 1) − 1, and k and ℓ are the unique integers such that k ≥ 0,
1 ≤ ℓ ≤ q−2 and d = k(q −2)+ℓ. This uniform bound was given in [29, Theorem 3.5] and it
is seen that this bound is in fact optimal by constructing an appropriate polynomial f.
If fp (d) = δ (d) for d ≥ 1, we say that I is a Geil–Carvalho ideal. For vanishing ideals
I I
over finite fields, this notion is essentially another way of saying that the bound of Eq. (4.1) is
optimal. Thefirstinterestingfamilyofidealswhereequality holdsisduetoGeil[12,Theorem2].
His result essentially shows that fp (d) = δ (d) for d ≥ 1 when ≺ is a graded lexicographical
I I
order and I is the homogenization of the vanishing ideal of the affine space As−1 over a finite
field K = F . Recently, Carvalho [4, Proposition 2.3] extended this result by replacing As−1
q
by a cartesian product of subsets of F . In this case the underlying Reed-Muller-type code is
q
called an affine cartesian code and an explicit formula for the minimum distance was first given
in [13, 21]. In a very recent paper, Bishnoi, Clark, Potukuchi, and Schmitt give another proof
of this formula [3, Theorem 5.2] using a result of Alon and Fu¨redi [1, Theorem 5].
Asthetwomostrelevantapplications ofourmainresulttoalgebraiccodingtheory, werecover
the formula for the minimum distance of an affine cartesian code given in [21, Theorem 3.8] and
[13, Proposition 5], and the fact that the homogenization of the corresponding vanishing ideal
is a Geil–Carvalho ideal [4] (see Corollary 4.4).
Then we present an extension of a result of Alon and Fu¨redi [1, Theorem 1]—in terms of the
regularity of a vanishing ideal—about coverings of the cube {0,1}n by affine hyperplanes, that
can be applied to any finite subset of a projective space whose vanishing ideal has a complete
intersection initial ideal (see Corollary 4.5 and Example 4.6).
Finally, using Macaulay2 [16], we exemplify how some of our results can be used in practice,
and show that the vanishing ideal of P2 over F is not Geil–Carvalho by computing all possible
2
initial ideals (see Example 4.8).
In Section 2 we introduce projective Reed-Muller-type codes and present some of the results
andterminologythatwillbeneededinthepaper. Forallunexplainedterminologyandadditional
information, we refer to [38] (for deeper advances on the knowledge of the degree), [7] (for the
theoryofGro¨bnerbases),[2,10,32](forcommutativealgebraandHilbertfunctions),and[23,34]
(for the theory of error-correcting codes and linear codes).
4 JOSE´ MART´INEZ-BERNAL,YURIKOPITONES,ANDRAFAELH.VILLARREAL
2. Preliminaries
In this section, we present some of the results that will be needed throughout the paper and
introduce some more notation. All results of this section are well-known.
Let S = K[t ,...,t ] = ⊕∞ S be a graded polynomial ring over a field K with the standard
1 s d=0 d
grading and let (0) 6= I ⊂ S be a graded ideal. The Hilbert function of S/I is:
H (d) := dim (S /I ), d= 0,1,2,...
I K d d
where I = I ∩S . By the dimension of I we mean the Krull dimension of S/I.
d d
Theorem 2.1. (Hilbert [32, p. 58]) Let I ⊂ S be a graded ideal of dimension k. Then there is
a polynomial h (x) ∈ Q[x] of degree k−1 such that h (d) = H (d) for d≫ 0.
I I I
The degree or multiplicity of S/I is the positive integer
(k−1)! lim H (d)/dk−1 if k ≥ 1,
I
deg(S/I) := d→∞
( dimK(S/I) if k = 0.
If f ∈ S, the quotient ideal of I with respect to f is given by (I: f) = {h ∈ S|hf ∈ I}. The
element f is called a zero-divisor of S/I—as an S-module—if there is 0 6= a ∈ S/I such that
fa = 0, and f is called regular on S/I otherwise. Notice that f is a zero-divisor of S/I if and
only if (I: f) 6= I. An associated prime of I is a prime ideal p of S of the form p = (I: f) for
some f in S.
Definition 2.2. The regularity of the Hilbert function of S/I, or simply the regularity of S/I,
denoted reg(S/I), is the least integer r ≥ 0 such that H (d) is equal to h (d) for d≥ r.
I I
If S/I is a Cohen–Macaulay ring of dimension 1, then reg(S/I) is precisely the Castelnuovo–
Mumford regularity of S/I in the sense of [10, p. 509]. This follows readily from a well-known
formula that relates the a-invariant, the Castelnuovo–Mumford regularity, and the depth of the
ring S/I (see for instance [36, Corollary B.4.1]).
Definition 2.3. An ideal I ⊂ S is called a complete intersection if there exist g ,...,g in S
1 r
such that I = (g ,...,g ), where r is the height of I.
1 r
Remark 2.4. LetI ⊂ S beagradedideal. (a)Anytwominimalsetsofhomogeneousgenerators
of I have the same cardinality. (b) I is a complete intersection if and only if I is minimally
generated by a homogeneous regular sequence with ht(I) elements, where ht(I) denotes the
height of I (see [19, Chapter 3]). (c) A monomial ideal I is a complete intersection if and only
if I is minimally generated by a regular sequence of monomials with ht(I) elements.
If I is a graded ideal of S, the Hilbert series of S/I, denoted F (x), is given by
I
∞
F (x) = H (d)xd, where x is a variable.
I I
d=0
X
Theorem 2.5. (Hilbert–Serre [32, p. 58], [35, p. 297]) Let I ⊂ S be a graded ideal of dimension
k. Then there is a unique polynomial h(x) ∈Z[x] such that
h(x)
F (x) = and h(1) > 0.
I (1−x)k
Remark 2.6. The leading coefficient of the Hilbert polynomial h (x) is equal to h(1)/(k−1)!.
I
Thus h(1) is equal to deg(S/I).
COMPLETE INTERSECTIONS 5
Lemma 2.7. [32, Corollary 3.3] If I ⊂ S is an ideal generated by homogeneous polynomials
f ,...,f , with r = ht(I) and δ = deg(f ), then
1 r i i
r 1−xδi
F (x) = i=1 and deg(S/I) = δ ···δ .
I (1−x)s 1 r
Q (cid:0) (cid:1)
The footprint of an ideal. Let ≺ be a monomial order on S and let (0) 6= I ⊂ S be an ideal.
If f is a non-zero polynomial in S, one can write
f = λ tα1 +···+λ tαr,
1 r
with λ ∈K∗ = K\{0} for all i and tα1 ≻ ··· ≻ tαr. The leading monomial tα1 of f is denoted
i
by in (f). The initial ideal of I, denoted by in (I), is the monomial ideal of S given by
≺ ≺
in (I) := ({in (f)|f ∈ I}).
≺ ≺
Amonomialta iscalled astandard monomial ofS/I, withrespectto≺,ifta is nottheleading
monomial of any polynomial in I, that is, ta is not in the ideal in (I). A polynomial f is called
≺
standard if f 6= 0 and f is a K-linear combination of standard monomials.
The set of standard monomials, denoted ∆ (I), is called the footprint of S/I or Gr¨obner
≺
´escalier of I. The image of ∆ (I), under the canonical map S 7→ S/I, x 7→ x, is a basis of
≺
S/I as a K-vector space. This is a classical result of Macaulay (for a modern approach see [7,
Chapter 5, Section 3, Proposition 4]). In particular, if I is graded, then H (d) is the number of
I
standard monomials of degree d.
A subset G = {g ,...,g } of I is called a Gr¨obner basis of I if
1 r
in (I) = (in (g ),...,in (g )).
≺ ≺ 1 ≺ r
Lemma 2.8. Let ≺ be a monomial order, let I ⊂ S be an ideal, and let f be a polynomial of S
of positive degree. If in (f) is regular on S/in (I), then f is regular on S/I.
≺ ≺
Proof. Let g be a polynomial of S such that gf ∈ I. It suffices to show that g ∈ I. Pick a
Gro¨bner basis g ,...,g of I. Then, by the division algorithm [7, Theorem 3, p. 63], we can
1 r
r
write g = a g +h, where h= 0 or h is a standard polynomial of S/I. We need only show
i=1 i i
that h = 0. If h 6= 0, then hf is in I and in (h)in (f) is in in (I). Therefore in (h) is in
≺ ≺ ≺ ≺
in (I), a cPontradiction. (cid:3)
≺
Remark 2.9. Given an integer d ≥ 1, there is a map in : F → M given by f 7→ in (f).
≺ ≺,d ≺,d ≺
This follows from Lemma 2.8. If I is a monomial ideal, then M ⊂ F .
≺,d ≺,d
Projective Reed-Muller-type codes. Let K = F be a finite field with q elements, let Ps−1
q
beaprojective spaceover K, andlet X beasubsetof Ps−1. Asusual, points of Ps−1 aredenoted
by [α], where0 6= α ∈ Ks. In this paragraph all results are valid if we assume that K is any field
and X is a finite subset of Ps−1, instead of assuming that K is finite. However, the interesting
case for coding theory is when K is finite.
The vanishing ideal of X, denoted I(X), is the ideal of S generated by the homogeneous
polynomials that vanishat all points of X. In this case the Hilbertfunction of S/I(X) is denoted
by H (d). Let P ,...,P be a set of representatives for the points of X with m = |X|. Fix a
X 1 m
degreed ≥ 1. Foreach ithereisf ∈ S suchthatf (P ) 6= 0. IndeedsupposeP = [(a ,...,a )],
i d i i i 1 s
there is at least one k in {1,...,s} such that a 6= 0. Setting f (t ,...,t ) = td one has that
k i 1 s k
f ∈ S and f (P )6= 0. There is a K-linear map:
i d i i
f(P ) f(P )
(2.1) ev : S = K[t ,...,t ] → K|X|, f 7→ 1 ,..., m .
d d 1 s d
f (P ) f (P )
(cid:18) 1 1 m m (cid:19)
6 JOSE´ MART´INEZ-BERNAL,YURIKOPITONES,ANDRAFAELH.VILLARREAL
The map ev is called an evaluation map. The image of S under ev , denoted by C (d), is
d d d X
called a projective Reed-Muller-type code of degree d over X [9]. It is also called an evaluation
code associated to X [14]. This type of codes have been studied using commutative algebra
methods and especially Hilbert functions, see [8, 15, 26, 31] and the references therein.
Definition 2.10. A linear code is a linear subspace of Km for some m. The basic parameters
of the linear code C (d) are its length |X|, dimension dim C (d), and minimum distance
X K X
δ (d) := min{kvk: 0 6= v ∈ C (d)},
X X
where kvk is the number of non-zero entries of v.
Lemma 2.11. [24, Lemma 2.13] (a) The map ev is well-defined, i.e., it is independent of
d
the set of representatives that we choose for the points of X. (b) The basic parameters of the
Reed-Muller-type code C (d) are independent of f ,...,f .
X 1 m
Thefollowing summarizesthewell-known relation between projective Reed-Muller-typecodes
and the theory of Hilbert functions. Notice that items (i) and (iv) follow directly from Eq. (2.1)
and item (iii), respectively.
Proposition 2.12. The following hold.
(i) H (d) = dim C (d) for d≥ 1.
X K X
(ii) [18, Lecture 13] deg(S/I(X)) = |X|.
(iii) (Singleton bound) 1 ≤ δ (d) ≤ |X|−H (d)+1 for d ≥1.
X X
(iv) δ (d) = 1 for d≥ reg(S/I(X)).
X
The next result gives an algebraic formulation of the minimum distance of a projective Reed-
Muller-type code in terms of the degree and the structure of the underlying vanishing ideal.
Theorem 2.13. [24, Theorem 4.7] If |X|≥ 2, then δ (d) = δ (d) ≥ 1 for d ≥ 1.
X I(X)
This result gives an algorithm, that can be implemented in CoCoA [6], Macaulay2 [16], or
Singular [17], to compute δ (d) for small values of q and s, where q is the cardinality of F and
X q
s is the number of variables of S (see the procedure of Example 4.6). Using SAGE [27] one can
also compute δ (d) by finding a generator matrix of C (d).
X X
As a direct consequence of Theorem 2.13 one has:
(2.2) δ (d) = deg(S/I(X))−max{|V (f)|: f 6≡ 0;f ∈ S },
I(X) X d
where V (f) is the zero set of f in X and f 6≡ 0 means that f does not vanish at all points of X.
X
The next lemma follows using the division algorithm [7] (cf. [11, Problem 1-17]).
Lemma 2.14. Let X be a finite subset of Ps−1, let [α] be a point in X, with α = (α ,...,α )
1 s
and α 6= 0 for some k, and let I be the vanishing ideal of [α]. Then I is a prime ideal,
k [α] [α]
I = ({α t −α t |k 6= i∈ {1,...,s}), deg(S/I )= 1,
[α] k i i k [α]
ht(I ) = s−1, and I(X) = I is the primary decomposition of I(X).
[α] [β]∈X [β]
Remark 2.15. IfXisafiniteTsetofprojectivepoints,thenS/I(X)isaCohen–Macaulayreduced
graded ring of dimension 1. This is very well-known, and it follows directly from Lemma 2.14.
In particular, the regularity of the Hilbert function of S/I(X) is the Castelnuovo–Mumford
regularity of S/I(X).
The next result classifies monomial vanishing ideals of finite sets in a projective space.
COMPLETE INTERSECTIONS 7
Proposition 2.16. Let X be a finite subset of Ps−1. The following are equivalent:
(a) I(X) is a monomial ideal.
(b) I(X)= ∩m p , where the p ’s are generated by s−1 variables.
i=1 i i
(c) X ⊂ {[e ],...,[e ]}, where e is the i-th unit vector.
1 s i
Proof. (a) ⇒ (b): By Remark 2.15, I(X) is a radical Cohen–Macaulay graded ideal of dimension
1. Hence, I(X) is an unmixed square-free monomial ideal of height s−1. Therefore, I(X) is
equal to ∩m p , where the p ’s are face ideals (i.e., ideals generated by variables) of height s−1.
i=1 i i
(b) ⇒ (c): Let X be the Zariski closure of X. As X is finite, one has X = X = V(I(X)) =
∪m V(p ). Thus it suffices to notice that V(p )= {[e ]} for some i .
i=1 i i ij j
(c) ⇒ (a): This follows from Lemma 2.14. (cid:3)
3. Complete intersections
LetS = K[t ,...,t ] = ⊕∞ S beapolynomial ringover a fieldK withthestandardgrading
1 s d=0 d
and s ≥ 2. In what follows by a monomial order ≺ we mean a graded monomial order in the
sense that ≺ is defined first by total degree [7].
Lemma 3.1. Let L ⊂ S be an ideal generated by monomials. If dim(S/L) = 1, then L is a
complete intersection if and only if, up to permutation of the variables t ,...,t , we can write
1 s
(i) L = (td2,...,tds) with 1 ≤ d ≤ d for i≥ 2, or
2 s i i+1
(ii) L = (td12,...,tdpp−1,tcpptpcp++11,tpd+p+22,...,tdss) for some p ≥ 1 such that 1 ≤ cp ≤ cp+1 and
1≤ d ≤ d for 2≤ i ≤ s−1, where d = c +c .
i i+1 p+1 p p+1
Proof. ⇒) Let tα1,...,tαs−1 be the minimal set of generators of L consisting of monomials.
These monomials form a regular sequence (see Remark 2.4). Hence tαi and tαj have no common
variables for i 6= j. Then, either all variables occur in tα1,...,tαs−1 and we are in case (ii), up
to permutation of the variables t1,...,ts, or there is one variable that is not in any of the tαi’s
and we are in case (i), up to permutation of the variables t ,...,t .
1 s
⇐) In both cases L is an ideal of height s−1 generated by s−1 elements, that is, L is a
complete intersection. (cid:3)
Proposition 3.2. [37, Propositions 3.1.33 and 5.1.11] Let A = R /I , B = R /I be two
1 1 2 2
standard graded algebras over a field K, where R = K[x], R = K[y] are polynomial rings in
1 2
disjoint sets of variables and I is an ideal of R . If R = K[x,y] and I =I R+I R, then
i i 1 2
(R /I )⊗ (R /I ) ≃ R/I and F(A⊗ B,x)= F(A,x)F(B,x),
1 1 K 2 2 K
where F(A,x) and F(B,x) are the Hilbert series of A and B, respectively.
Lemma 3.3. Let L be the ideal of S generated by td2,...,tds. If ta = ta1tar ···tas, r ≥ 2,
2 s 1 r s
a ≥ 1, and a ≤ d −1 for i≥ r, then
r i i
deg(S/(L,ta)) = deg(S/(L,tar ···tas)) = d ···d −d ···d (d −a )···(d −a ),
r s 2 s 2 r−1 r r s s
where a = 0 if 2 ≤ i < r.
i
Proof. In what follows we will use the fact that Hilbert functions and Hilbert series are additive
on short exact sequences [11, Chapter 2, Proposition 7]. If a ≥ 1, then taking Hilbert functions
1
in the exact sequence
0 −→ S/(L,tar ···tas)[−a ]−ta1→1 S/(L,ta)−→ S/(L,ta1) −→ 0,
r s 1 1
8 JOSE´ MART´INEZ-BERNAL,YURIKOPITONES,ANDRAFAELH.VILLARREAL
and noticing that dim(S/(L,ta1)) = 0, the first equality follows. Thus we may assume that ta
1
has the form ta = tar ···tas and a = 0 for i< r.
r s i
We proceed by induction on s ≥ 2. Assume s = 2. Then r = 2, ta = ta2, (L,ta) = (ta2),
2 2
and the degree of S/(L,ta) is a , as required. Assume s ≥ 3. If a = 0 for i > r, then
2 i
(L,ta) = (L,tar) is a complete intersection and the required formula follows from Lemma 2.7.
r
Thus we may assume that a ≥ 1 for some i > r. There is an exact sequence
i
(3.1) 0 −→ S/(td2,...,tdr−1,tdr−ar,tdr+1,...,tds,tar+1···tas)[−a ] −tar→r
2 r−1 r r+1 s r+1 s r
S/(L,ta) −→ S/(td2,...,tdr−1,tar,tdr+1,...,tds)−→ 0.
2 r−1 r r+1 s
Noticethattheringontherightisacompleteintersectionandtheringontheleftisisomorphic
to the tensor product
(3.2) K[t ,...,t ]/(td2,...,tdr−1,tdr−ar)⊗ K[t ,t ,...,t ]/(tdr+1,...,tds,tar+1···tas).
2 r 2 r−1 r K 1 r+1 s r+1 s r+1 s
Hence, taking Hilbert series in Eq. (3.1), and applying Lemma 2.7, Theorem 2.5, and Propo-
sition 3.2, we get that the Hilbert series of S/(L,ta) can be written as
xar(1−xd2)···(1−xdr−1)(1−xdr−ar) g(x)
F(S/(L,ta),x) = +
(1−x)r−1 (1−x)
(1−xd2)···(1−xdr−1)(1−xar)(1−xdr+1)···(1−xds)
,
(1−x)s
where g(x)/(1−x) is the Hilbert series of the second ring in the tensor product of Eq. (3.2) and
g(1) is its degree (see Remark 2.6). By induction hypothesis
g(1) = d ···d −(d −a )···(d −a ).
r+1 s r+1 r+1 s s
Therefore,writingF(S/(L,ta),x) = h(x)/(1−x) withh(x) ∈ Z[x]andh(1) > 0, andrecalling
that h(1) is the degree of S/(L,ta), we get
deg(S/(L,ta)) = h(1) = d ···d (d −a )g(1)+d ···d a d ···d
2 r−1 r r 2 r−1 r r+1 s
= d ···d −(d −a )···(d −a ). ✷
2 s 2 2 s s
Lemma 3.4. (A) Let L be the ideal of S generated by td12,...,tdp−p1,tcpptpcp++11,tpd+p+22,...,tdss, where
p ≥ 1, 1 ≤ c ≤ c and d ≥ 1 for all i. If ta = ta1···tas is not in L, d = c +c , and
p p+1 i 1 s p+1 p p+1
a ≥ 1 for some i, then the degree of S/(L,ta) is equal to
i
p−1 s
(i) d ···d −(c −a ) (d −a ) (d −a ) if a ≥ c ;
2 s p+1 p+1 i+1 i i i p p
i=1 i=p+2
Y Y
p−1 s
(ii.1) d ···d −(c −a ) (d −a ) (d −a ) if a < c , a ≥ c ;
2 s p p i+1 i i i p p p+1 p+1
i=1 i=p+2
Y Y
p−1 s
(ii.2) d ···d −(d −a −a ) (d −a ) (d −a ) if a < c , a < c .
2 s p+1 p p+1 i+1 i i i p p p+1 p+1
i=1 i=p+2
Y Y
(B) Let I be a graded ideal such that L = in (I). If 0 ≤ k ≤ s−2, ℓ are integers such that
≺
d= k+1(d −1)+ℓ and 1 ≤ ℓ ≤ d −1, then fp (d) ≤ (d −ℓ)d ···d .
i=2 i k+2 I k+2 k+3 s
P
COMPLETE INTERSECTIONS 9
Proof. (A) Case (i): Assume a ≥ c . If a = 0 for i 6= p, then ta = tap, and by the first equality
p p i p
of Lemma 3.3, and using Lemma 2.7, we get
deg(S/(L,ta)) = deg(S/(td2,...,tdp ,tap,tdp+2,...,tds,tcptcp+1))
1 p−1 p p+2 s p p+1
= deg(S/(td2,...,tdp ,tcp,tdp+2,...,tds)) = d ···d c d ···d
1 p−1 p p+2 s 2 p p p+2 s
= d ···d (d −c )d ···d = d ···d −c d ···d d ···d ,
2 p p+1 p+1 p+2 s 2 s p+1 2 p p+2 s
as required. We may now assume that a ≥ 1 for some i 6= p. As ta ∈/ L and a ≥ c , one has
i p p
a < d for i = 1,...,p−1, a < c , and a < d for i = p+2,...,s. Therefore from the
i i+1 p+1 p+1 i i
exact sequence
0 −→ S/(td2,...,tdp ,tcp+1,tdp+2,...,tds,ta1···tap−1tap−cptap+1···tas)[−c ] −tcp→p
1 p−1 p+1 p+2 s 1 p−1 p p+1 s p
S/(L,ta)−→ S/(td2,...,tdp ,tcp,tdp+2,...,tds) −→ 0,
1 p−1 p p+2 s
and using Lemmas 3.3 and 2.7, the required equality follows.
Case (ii): Assume a < c . If a = 0 for i 6= p, then ta = tap and 0 = a < c . Hence, by
p p i p p+1 p+1
Lemma 2.7, we get
deg(S/(L,ta)) = deg(S/(td2,...,tdp ,tap,tdp+2,...,tds))
1 p−1 p p+2 s
= d ···d a d ···d
2 p p p+2 s
= d ···d −(d −a )d ···d d ···d ,
2 s p+1 p 2 p p+2 s
as required. We may now assume that a ≥ 1 for some i6= p. Consider the exact sequence
i
0 −→ S/(td2,...,tdp ,tcp+1,tdp+2,...,tds,ta1···tap−1tap+1···tas)[−c ]−tcp→p
1 p−1 p+1 p+2 s 1 p−1 p+1 s p
(3.3) S/(L,ta) −→ S/(td2,...,tdp ,tcp,tdp+2,...,tds,ta1···tas) −→ 0.
1 p−1 p p+2 s 1 s
Subcase (ii.1): Assume a ≥ c . As ta ∈/ L, in our situation, one has a < d for
p+1 p+1 i i+1
i = 1,...,p−1, a < c , and a < d for i = p+2,...,s. If a = 0 for i 6= p+1, then taking
p p i i i
Hilbert series in Eq. (3.3), and noticing that the ring on the right has dimension 0, we get
deg(S/(L,ta)) = d ···d c d ···d
2 p p+1 p+2 s
= d ···d −c d ···d d ···d ,
2 s p 2 p p+2 s
as required. Thus we may now assume that a ≥ 1 for some i 6= p+1. Taking Hilbert series in
i
Eq. (3.3), and using Lemma 2.7, we obtain
deg(S/(L,ta)) = d ···d c d ···d +
2 p p+1 p+2 s
deg(S/(td2,...,tdp ,tcp,tdp+2,...,tds,ta1···tas)).
1 p−1 p p+2 s 1 s
Therefore, using Lemma 3.3, the required equality follows.
Subcase (ii.2): Assume a < c . If a = 0 for i6= p+1, taking Hilbert series in Eq. (3.3),
p+1 p+1 i
and noticing that the ring on the right has dimension 0, by Lemma 3.3, we get
deg(S/(L,ta)) = d ···d c d ···d −d ···d (c −a )d ···d
2 p p+1 p+2 s 2 p p+1 p+1 p+2 s
= d ···d a d ···d
2 p p+1 p+2 s
= d ···d −(d −a )d ···d d ···d ,
2 s p+1 p+1 2 p p+2 s
as required. Thus we may now assume that a ≥ 1 for some i 6= p+1. Taking Hilbert series in
i
Eq. (3.3), and applying Lemma 3.3 to the ends of Eq. (3.3), the required equality follows.
10 JOSE´ MART´INEZ-BERNAL,YURIKOPITONES,ANDRAFAELH.VILLARREAL
(B) It suffices to find a monomial tb in M such that
≺,d
(3.4) deg(S/(in (I),tb)) = (d −ℓ)d ···d .
≺ k+2 k+3 s
There are five cases to consider:
td2−1···tdp−1tcptcp+1−1tdp+2−1···tdk+1−1tℓ if k ≥ p+1,
1 p−1 p p+1 p+2 k+1 k+2
 td12−1···tdpp−−11tcpptpcp++11−1tℓp+2 if k = p,
tb =  td12−1···tdkk+1−1tℓk+1 if k ≤ p−2,
 td12−1···tdpp−−11tcpptpℓ−+c1p if k = p−1 and ℓ ≥ cp,
td2−1···tdp−1tℓ if k = p−1 and ℓ < c .
 1 p−1 p p


In each case,by the formulas for the degree of part (A), we get the equality of Eq. (3.4). (cid:3)
An ideal I ⊂ S is called unmixed if all its associated primes have the same height and I is
called radical if I is equal to its radical. The radical of I is denoted by rad(I).
Lemma 3.5. [24, Lemma 4.1] Let I ⊂ S be an unmixed graded ideal and let ≺ be a monomial
order. If f ∈ S is homogeneous and (I: f)6= I, then
deg(S/(I,f)) ≤ deg(S/(in (I),in (f))) ≤ deg(S/I),
≺ ≺
and deg(S/(I,f)) < deg(S/I) if I is an unmixed radical ideal and f ∈/ I.
Remark 3.6. Let I ⊂ S be an unmixed graded ideal of dimension 1. If f ∈ S , then (I: f)= I
d
if and only if dim(S/(I,f)) = 0. In this case deg(S/(I,f)) could be greater than deg(S/I).
Lemma 3.7. [24, Lemma 3.2] Let X be a finite subset of Ps−1 over a field K and let I(X)⊂ S
be its graded vanishing ideal. If 0 6= f ∈ S is homogeneous, then the number of zeros of f in X
is given by
degS/(I(X),f) if (I(X): f)6= I(X),
|V (f)| =
X 0 if (I(X): f)= I(X).
(cid:26)
Corollary 3.8. Let I = I(X) be the vanishing ideal of a finite set X of projective points, let
f ∈F , and in (f) = ta1···tas. If in (I) is generated by td2,...,tds, then there is r ≥ 2 such
≺,d ≺ 1 s ≺ 2 s
that a ≥ 1, a ≤ d −1 for i ≥ r, a = 0 if 2 ≤ i < r, and
r i i i
|V (f)|≤ d ···d −(d −a )···(d −a ).
X 2 s 2 2 s s
Proof. As f is a zero-divisor of S/I, by Lemma 2.8, ta = in (f) is a zero-divisor of S/in (I).
≺ ≺
Hence, there is r ≥ 2 such that a ≥ 1 and a = 0 if 2 ≤ i < r. Using that ta is a standard
r i
monomial of S/I, we get that a ≤ d −1 for i ≥ r. Therefore, using Lemma 3.7 together with
i i
Lemmas 3.3 and 3.5, we get
|V (f)| = deg(S/(I(X),f)) ≤ deg(S/(in (I(X)),in (f)))
X ≺ ≺
= d ···d −(d −a )···(d −a ). ✷
2 s 2 2 s s
Lemma 3.9. Let I be a graded ideal and let ≺ be a monomial order. Then the minimum
distance function δ is independent of ≺.
I
Proof. Fix a positive integer d. Let F be the set of all homogeneous zero-divisors of S/I not
d
in I of degree d and let f be an element of F . Pick a Gro¨bner basis g ,...,g of I. Then,
d 1 r
r
by the division algorithm [7, Theorem 3, p. 63], we can write f = a g +h, where h is a
i=1 i i
P