Table Of ContentFATTENING UP WARNING’S SECOND THEOREM
PETEL.CLARK
Abstract. WepresentageneralizationofWarning’sSecondTheoremtopoly-
nomialsystemsoverafinitelocalprincipalringwithsuitablyrestrictedinput
andoutputvariables. ThisgeneralizesarecentresultwithForrowandSchmitt
(andgivesanewproofofthatresult). Applicationstoadditivegrouptheory,
graphtheoryandpolynomialinterpolationarepursuedindetail.
Contents
1. Introduction 1
1.1. Notation and Terminology 1
1.2. Prior Results 2
1.3. The Main Theorem 3
1.4. Comparison With Theorem 1.2 6
1.5. Applications of the Main Theorem 6
1.6. Acknowledgments 7
2. Proof of the Main Theorem 7
2.1. A Generalized Alon-Friedman-Kalai Lemma 7
2.2. Alon-Fu¨redi Over a Ring 8
2.3. Proof of the Main Theorem 10
3. Applications 11
3.1. Hypergraphs 11
3.2. Fat Davenport Constants 12
3.3. Divisible Subgraphs 16
3.4. Divisibility in Weighted Graphs 18
3.5. Polynomial Interpolation With Fat Targets 19
References 20
1. Introduction
1.1. Notation and Terminology.
Let n,a ,...,a ∈Z+ and let 1≤N ≤(cid:80)n a . As in [CFS14, §2.1], we put
1 n i=1 i
(cid:40)
1 N <n
m(a ,...,a ;N)= ;
1 n min(cid:81)n y n≤N ≤(cid:80)n a
i=1 i i=1 i
the minimum is over (y ,...,y )∈Zn with y ∈[1,a ] for all i and (cid:80)n y =N.
1 n i i i=1 i
Let R be a ring, B ⊂ R a subset, I an ideal of R, and x ∈ R. We write “x ∈ B
1
2 PETEL.CLARK
(mod I)” to mean that there is b∈B such that x−b∈I.
LetRbearing. Asin[Cl14],wesayasubsetA⊂RsatisfiesCondition (F)(resp.
Condition (D)) if A is nonempty, finite and for any distinct elements x,y ∈ A,
x−y is a unit in R (resp. is not a zero-divisor in R).
1.2. Prior Results.
We begin with the results of Chevalley and Warning.
Theorem 1.1. Let n,r,d ,...,d ∈Z+ with d:=d +...+d <n. For 1≤i≤r,
1 r 1 r
let f (t ,...,t )∈F [t]=F [t ,...,t ] be a polynomial of degree d . Let
i 1 n q q 1 n i
Z =Z(f ,...,f )={x∈Fn |f (x)=...=f (x)=0}.
1 r q 1 r
a) (Chevalley’s Theorem [Ch35]) We have #Z =0 or #Z ≥2.
b) (Warning’s Theorem [Wa35]) We have #Z ≡0 (mod p).
c) (Warning’s Second Theorem [Wa35]) We have #Z =0 or #Z ≥qn−d.
Chevalley’s proof of Theorem 1.1a) can be easily modified to yield Theorem 1.1b).
Warning’s real contribution was Theorem 1.1c), a result which has, I feel, been too
little appreciated. It is sharp in the following strong sense: for any d ,...,d ∈ N
1 r
with d := d +...+d < n, there are f ,...,f ∈ F [t] with degf = d for all
1 r 1 r q i i
1≤i≤r such that #Z(f ,...,f )=qn−d. One can build such examples by com-
1 r
biningnormformsassociatedtofieldextensionsF /F andlinearpolynomials. On
qa q
the other hand, although in these examples the equations are generally nonlinear,
the solution sets are still affine subspaces. In [HB11], Heath-Brown showed that
under the hypotheses of Theorem 1.1c), when Z is nonempty and is not an affine
subspace of Fn one always has #Z >qn−d, and in fact #Z ≥2qn−d for all q ≥4.
q
Apart from [HB11] there had been little further exploration of Theorem 1.1c) until
[CFS14], in which A. Forrow, J.R. Schmitt and I established the following result.
Theorem 1.2. (Restricted Variable Warning’s Second Theorem [CFS14]) Let K
beanumberfieldwithringofintegersR,letpbeanonzeroprimeidealofR,andlet
q =p(cid:96) be the prime power such that R/p∼=F . Let A ,...,A be nonempty subsets
q 1 n
of R such that for each i, the elements of A are pairwise incongruent modulo p,
i
and put A=(cid:81)n A . Let r,v ,...,v ∈Z+. Let P ,...,P ∈R[t ,...,t ]. Let
i=1 i 1 r 1 r 1 n
Z ={x∈A|P (x)≡0 (mod pvj) ∀1≤j ≤r}, z =#Z .
A j A A
(cid:16) (cid:17)
ThenzA =0orzA ≥m #A1,...,#An;#A1+...+#An−(cid:80)rj=1(qvj −1)deg(Pj) .
This generalizes Theorem 1.1c) in two directions: first, instead of working over
finite fields, we work modulo powers of a prime ideal in the ring of integers of a
number field. In the case K = Q we are studying systems of congruence modulo
(varying) powers of a (fixed) prime p. Second, we study solutions in which each
variableisindependentlyrestrictedtoafinitesubsetofZ satisfyingthecondition
K
that no two distinct elements are congruent modulo p.
These extensions appear already in work of Schanuel [Sc74], Baker-Schmidt
[BS80], Schauz [Sc08], Wilson [Wi06] and Brink [Br11]. They are largely moti-
vated by applications to combinatorics. For combinatorial applications we work
FATTENING UP WARNING’S SECOND THEOREM 3
over K = Q and get congruences modulo powers of p. The most classical applica-
tions concern the case in which each variable is restricted to take values 0 and 1.
More recently there has been a surge of interest in more general subsets A : this
i
yields weighted analogues of the more classical combinatorial problems.
ThepreviousworksusedeitheradhocmethodsorAlon’sCombinatorialNullstel-
lensatzandyieldednonuniqueness theorems: resultswithconclusion“therecannot
beexactlyonesolution”. ToproveTheorem1.2weinsteadappliedtheAlon-Fu¨redi
Theorem, which yields a lower bound on the number of solutions in terms of the
quantity m(a ,...,a ;N). To collapse this type of result to a nonuniqueness theo-
1 n
rem one simply uses the Pigeonhole Principle
(1) m(a ,...,a ;N)≥2 ⇐⇒ N >n.
1 n
Applying (1) to Theorem 1.2, one recovers a result of Brink.
Corollary 1.3. (Brink [Br11]) Let K be a number field with ring of integers R,
let p be a nonzero prime ideal of R, and let q = p(cid:96) be the prime power such that
R/p∼=F . Let P (t ,...,t ),...,P (t ,...,t )∈R[t ,...,t ], let v ,...,v ∈Z+,
q 1 1 n r 1 n 1 n 1 r
and let A ,...,A be nonempty subsets of R such that for each i, the elements of
1 n
A are pairwise incongruent modulo p, and put A=(cid:81)n A . Let
i i=1 i
Z ={x∈A|P (x)≡0 (mod pvj) ∀1≤j ≤r}, z =#Z .
A j A A
If (cid:80)rj=1(qvj −1)deg(Pj)<(cid:80)ni=1(#Ai−1), then zA (cid:54)=1.
The case of K = Q was independently (in fact, earlier) established by U. Schauz
and R. Wilson, so we call this result the Schauz-Wilson-Brink Theorem. If we
further specialize to A ={0,1} for all i we recover Schanuel’s Theorem.
i
1.3. The Main Theorem.
For the convenience of readers who are primarily interested in combinatorial appli-
cations, we state the main result of this paper first in a special case.
Theorem 1.4. Let p be a prime, let n,r,v ∈Z+, and for 1≤i≤r, let 1≤v ≤v.
j
Let A ,...,A ,B ,...,B ⊂ Z/pvZ be nonempty subsets each having the property
1 n 1 r
that no two distinct elements are congruent modulo p. Let f ,...,f ∈Z/pvZ[t]=
1 r
Z/pvZ[t ,...,t ]. Let
1 n
n
(cid:89)
ZB ={x∈ A |∀1≤j ≤r, f (x)∈B (mod pvj)}.
A i j j
i=1
Then ZB =∅ or
A
n r
(cid:88) (cid:88)
#ZAB ≥m#A1,...,#An; #Ai− (pvj −#Bj)degfj.
i=1 j=1
Corollary 1.5. Maintain the setup of Theorem 1.4.
a) If A ={0,1} for all 1≤i≤n, then ZB =∅ or
i A
#ZAB ≥2n−(cid:80)rj=1(pvj−#Bj)deg(fj).
b) If A ={0,1} for all i and f (0)=0∈B for all j, there is 0(cid:54)=x∈ZB if
i j j A
r
(cid:88)
n> (pvj −#B )deg(f ).
j j
j=1
4 PETEL.CLARK
Proof. Applying Theorem 1.4 in the case A = ... = A = {0,1} and using the
1 n
fact that for any 0 ≤ k ≤ n, we have m(2,...,2;2n−k) = 2n−k [CFS14, Lemma
2.2c)], we get part a). Combining with (1) we get part b). (cid:3)
If in Corollary 1.5b) we further require that all the polynomials are linear, we re-
cover a result of Alon-Friedland-Kalai [AFK84, Thm. A.1]. For some (not all)
combinatorial applications linear polynomials are sufficient, and A ={0,1} corre-
i
sponds to the “unweighted” combinatorial setup. In this setting we see that the
advantage of Corollary 1.5a) over part b) is directly analogous to that of Theorem
1.2 over Brink’s Theorem, namely a quantitative refinement of Alon-Fu¨redi type.
InfactthisgivesanaccurateglimpseofourmethodofproofoftheMainTheorem:
we will establish and apply suitably generalized versions of a valuation-theoretic
lemma of Alon-Friedland-Kalai and of the Alon-Fu¨redi Theorem.
To state the full version of the Main Theorem we need some algebraic prelimi-
naries. A principal ring is a commutative ring in which every ideal is principal.
A ring is local if it has exactly one maximal ideal. Let (r,p) be a local principal
ring with maximal ideal p = (π). By Nakayama’s Lemma, (cid:84) pi = (0), so for
i≥0
every nonzero x∈r, there is a unique i∈N such that x∈pi\pi+1, so x=πiy and
y is a unit in r, so (x)=(πi)=pi. Thus every nonzero ideal of r is of the form pi
for some i∈N. There are two possibilities:
(i) For all a∈Z+, pa (cid:54)=0. Then r is a DVR.
(ii)Thereisapositiveintegerv,thelengthofr,suchthatpv−1 (cid:54)=(0)andpv =(0).
If r is moreover finite then (ii) must hold. Thus in any (nonzero) finite princi-
pal ring (r,p) there is a positive integer v such that the ideals of r are
r=p0 (cid:41)p(cid:41)p1 (cid:41)...(cid:41)pv =(0).
Theorem1.6. Let(r,p)beafinitelocalringoflengthvandwithresiduefieldr/p∼=
F . Let n,r ∈Z+, and for 1≤j ≤r, let 1≤v ≤v. Let a ,...,a ,b ,...,b ⊂r
q j 1 n 1 r
be nonempty subsets each having the property that no two distinct elements are
congruent modulo p. Let f ,...,f ∈r[t]=r[t ,...,t ]. Let
1 r 1 n
n
(cid:89)
za,b ={x∈ ai |∀1≤j ≤r, fj(x)∈bj (mod pvj)}.
i=1
Then z =∅ or
a,b
n r
(cid:88) (cid:88)
#za,b ≥m#a1,...,#an; #ai− (qvj −#bj)degfj.
i=1 j=1
Consider the following variant of Theorem 1.6.
Theorem1.7. LetRbeaDedekinddomainwithmaximalidealpandfiniteresidue
field R/p ∼= F . Let n,r,v ,...,v ∈ Z+. Let A ,...,A ,B ,...,B ⊂ R be
q 1 r 1 n 1 r
nonempty subsets each having the property that no two distinct elements are con-
gruent modulo p. Let r,v ,...,v ∈Z+. Let f ,...,f ∈R[t ,...,t ]. Put
1 r 1 r 1 n
(cid:40) n (cid:41)
(cid:89)
ZB = x∈ A |∀1≤j ≤r,f (x)∈B (mod pvj) .
A i j j
i=1
FATTENING UP WARNING’S SECOND THEOREM 5
Then #ZB =0 or
A
n r
(cid:88) (cid:88)
#ZAB ≥m#A1,...,#An; #Ai− (qvj −#Bj)deg(fj).
i=1 j=1
Remark 1.8. By replacing R with R , one immediately reduces the statement of
p
Theorem 1.7 to the case in which R is a discrete valuation ring. In this setting, the
hypothesis on the A and B is simply Condition (F).
i j
Proposition 1.9. Theorems 1.6 and 1.7 are equivalent.
Proof. Theorem 1.6 =⇒ Theorem 1.7: let r = R/pv and let q : R → r be the
quotient map. For 1 ≤ i ≤ n, let a = q(A ); for 1 ≤ j ≤ r, let f = q(f ) and
i i j j
b =q(B ). Then degf ≤degf . The hypothesis that no two distinct elements of
j j j j
any one of these sets are congruent modulo p ensures #a =#A and #b =#B .
i i j j
Applying Theorem 1.6 to r,a ,...,a ,b ,...,b ,v ,...,v ,f ,...,f gives
1 n 1 r 1 r 1 r
n r
(cid:88) (cid:88)
#ZAB =#za,b ≥m#a1,...,#an; #ai− (qvj −#bj)deg(fj)
i=1 j=1
n r
(cid:88) (cid:88)
≥m#A1,...,#An; #Ai− (qvj −#Bj)deg(fj).
i=1 j=1
Theorem 1.7 =⇒ Theorem 1.6: the Cohen structure theorems imply that an
Artinian local principal ring is a quotient of a Dedekind domain (equivalently, of a
DVR) [Hu68, Cor. 11]. Thus we may write r = R/pv for a Dedekind domain R.
WemayliftA ,...,A ,B ,...,B ,f ,...,f fromrtoR soastopreservethesizes
1 n 1 r 1 r
of the sets and the degrees of the polynomials. Apply Theorem 1.7. (cid:3)
Example 1.10. Let r be a finite commutative ring, and let A⊂r satisfy Condition
(D). For 1 ≤ i ≤ r, let f ∈ r[t ] be a univariate polynomial of degree d ≥ 0, and
j 1 i
let B ,...,B ⊂r be finite and nonempty. Let
1 r
zB ={x∈A|f (x)∈B ,...,f (x)∈B }.
A 1 1 r r
Suppose first that d ≥ 1 for all j. Then for each y ∈ r, the polynomial f −y
j j
also has degree d , and because A satisfies Condition (D), there are at most deg(f )
j j
elements of A such that f (x) = y. So there are at most (#r − #B )(degf )
j j j
elements x∈A such that f (x)∈/ B and thus
j j
r
(cid:88)
#zB ≥#A− (#r−#B )(degf ).
A j j
j=1
Now suppose that some f is constant. Then: if the constant value lies in B then
j j
f (A)⊂B , whereas if the constant value does not lie in B then zB =∅.
j j j A
This establishes a stronger result than Theorem 1.6 when n = 1. In particular:
the finite ring r need not be local and principal, the target sets B ,...,B need not
1 r
satisfy Condition (F) but rather may be arbitrary nonempty subsets, and we do not
need to separately allow zB =∅ if each polynomial has positive degree.
A
6 PETEL.CLARK
1.4. Comparison With Theorem 1.2.
Theorem 1.2 is the special case of Theorem 1.7 obtained by taking R = Z and
K
B = {0} for all j. Thus on the face of it Theorem 1.7 is a twofold generalization
j
ofTheorem1.2: inplaceofZ wemaytakeanypair(R,p)withR aDedekinddo-
K
main and p a prime ideal such that R/p is a finite field; and in place of polynomial
congruenceswearestudyingpolynomialsystemswithrestricted output setsB .
j
The first generalization turns out not to be an essential one. Theorem 1.6 shows
that the result can be phrased in terms of finite, local principal rings. But every
finite local principal ring is isomorphic to Z /pv for some prime ideal p in the
K
ring of integers of a number field K. This is due to A.A. Neˇcaev [Ne71]. A more
streamlined proof appears in [BC15].
Example 1.11. Consider r = F [t]/(t2): it is a finite, local principal ring with
p
residue cardinality p and length 2. Further, it is a commutative F -algebra of di-
p
mension 2 which is not reduced: i.e., it has nonzero nilpotent elements, and this
√
latterdescriptioncharacterizesruptoisomorphism. SoletK =Q( p)andletpbe
theuniqueprimeidealofZ dividingp. TheringZ /pZ =Z /p2 isalsoacom-
K K K K
mutativeF -algebraofdimension2whichisnotreduced,sor=F [t]/(t2)∼=Z /p2.
p p K
NeverthelessitisnaturaltothinkintermsofDedekinddomains,andswitchingfrom
one Dedekind domain to another seems artificial. The proof of Theorem 1.2 uses
thefactthatZ hascharacteristiczeroinanessentialway: akeytechnicaltoolwas
K
the use of Schanuel-Brink operators to replace a congruence modulo pv in Z
K
withasystemofcongruencesmodulop. AsSchanuelpointedout,thisconstruction
is morally about Witt vectors and thus particular to unequal characteristic. Our
proof of Theorem 1.7 does not reduce to the number field case but works directly
in any Dedekind domain. Applied to R = Z with B = {0} for all j, it gives a
K j
new proof of Theorem 1.2. This new approach feels more transparent and more
fundamental, and we hope that it will be more amenable to further generalization.
1.5. Applications of the Main Theorem.
The generalization from polynomial congruences to polynomial congruences with
restrictedoutputsallowsawiderangeofapplications. Aswementionedin[CFS14],
wheneveronehasacombinatorialexistencetheoremprovedviatheSchauz-Wilson-
Brink Theorem (or an argument that can be viewed as a special case theroef) one
can apply instead Theorem 1.2 to get a lower bound on the number of solutions.
Moreover, most applications of the Schauz-Wilson-Brink Theorem include a “ho-
mogeneity” condition which ensures the existence of a trivial solution. Theorem
1.2 applies also in the “inhomogeneous case”.
All of these applications can be generalized by allowing restricted outputs. In
[CFS14] we gave three combinatorial applications of Theorem 1.2: to hypergraphs,
togeneralizationsoftheErd˝os-Ginzburg-ZivTheorem, andtoweightedDavenport
constants. Intheformertwocases, wecan(andshall)immediatelyapplytheMain
Theorem to get stronger results. We include the proof of the hypergraph theorem
to showcase the use of nonlinear polynomials. We omit the proof of the EGZ-type
theorem: the proof given in [CFS14] of the special case adapts immediately.
FATTENING UP WARNING’S SECOND THEOREM 7
Our Main Theorem leads to a generalization of the weighted Davenport constant
that we call the fat Davenport constant. This seems to be an interesting ob-
ject of study in its own right and we include some general discussion. The fat
Davenport constant can also be used to extend results of Alon-Friedland-Kalai on
divisible subgraphs. This is a privileged application: the restricted output aspect of
the Main Theorem was directly inspired by [AFK84].
One reason that the combinatorial applications are interesting is that the upper
boundstheygiveare–intheunweighted,zero-outputcase–accompaniedbylower
bounds coming from elementary combinatorial constructions, which has the effect
of showing sharpness in Schanuel’s Theorem in certain cases. It is an interesting
challenge,notmethere,tofindothertypesofrestrictedinputsetsA andrestricted
i
output sets B illustrating sharpness in our generalized theorems.
j
Finally, we give an application of the Main Theorem to polynomial interpolation
with fat targets. As a special case we will deduce a generalization of a Theorem of
Troi-Zannier [TZ97] which was proved by them via more combinatorial means.
1.6. Acknowledgments.
Thanks to Dino Lorenzini, Paul Pollack and Lori D. Watson for helpful discus-
sions. ThankstoBobRumelyforsuggestingtheterminologyof“fattargets”. Iam
deeply indebted to John R. Schmitt for introducing me to this rich circle of ideas,
for many helpful remarks, and for Example 3.2.
2. Proof of the Main Theorem
2.1. A Generalized Alon-Friedman-Kalai Lemma.
The following result is a generalization of [AFK84, Lemma A.3].
Lemma 2.1. Let R be a discrete valuation ring, with maximal ideal p = (π) and
finite residue field R/p ∼= F . Let v ∈ Z+, and let S(v) be a set of coset represen-
q
tatives for pv in R. Let T ⊂ R satisfy Condition (F): no two distinct elements of
T are congruent modulo p, and let T be the image of T in R/pv. Let x∈R. Put
(cid:89)
P(x,v,T)= (x−y)
y∈S(v)\T
and
v−1
c(v)=(cid:88)(cid:0)qi−1(cid:1).
i=1
Then we have:
(2) ord P(x,v,T)≥c(v),
p
(3) ord P(x,v,T)=c(v) ⇐⇒ there is y ∈T such that ord (x−y)≥v.
p p
(cid:81)
Proof. Put P = y.
0 y∈S(v)\{pv}
Step 1: Suppose T ={y } and ord (x−y )≥v. As y runs through S(v)\T, x−y
0 p 0
8 PETEL.CLARK
runs through a set of representatives of the nonzero cosets of pv in R, and since if
x≡y (cid:54)≡0 (mod pv) then ord x=ord y, we have
p p
ord P(x,v,T)=ord P
p p 0
v−1 v−1
(cid:88) (cid:88)
= i·(#{x∈(pi∩S(v))\(pi+1∩S(v))})= i·(qv−i−qv−i−1)
i=0 i=0
=(qv−1−qv−2)+2(qv−2−qv−3)+3(qv−3−qv−4)+...+(v−1)(q−1)
v−1
(cid:88)
=(qv−1+qv−2+...+1)−(v−1)= (qi−1)=c(v).
i=1
Step 2: Suppose T ={y } and ord (x−y )<v. Then there is a unique y ∈S(v)
0 p 0 1
(cid:16) (cid:17)
with x≡y (mod pv), and y (cid:54)=y . Then we have P(x,v,T)=P x−y1 , so
1 1 0 0 x−y0
ord P(x,v,T)=c(v)+ord (x−y )−ord (x−y )>c(v).
p p 1 p 0
Step3: Suppose#T >1. ThenP(x,v,T)isobtainedfromomittingfactorsfroma
productconsideredinStep1orStep2. BecausenotwoelementsofT arecongruent
modulo p, the number of y ∈T such that ord (x−y)≥1 is either 0 or 1, and thus
p
P(x,v,T) can be obtained from the product in Step 1 or Step 2 by omitting only
factors of zero p-adic valuation. So ord P(x,v,T) ≥ c(v), and strict inequality
p
holds precisely when there is some y ∈S(v)\T with ord (x−y)≥v. (cid:3)
p
2.2. Alon-Fu¨redi Over a Ring.
The aim of this section is to prove the following result.
Theorem 2.2. (Alon-Fu¨redi Over a Ring) Let R be a ring, let A ,...,A ⊂ R
1 n
satisfying Condition (D). Put A = (cid:81)n A and a = #A for all 1 ≤ i ≤ n. Let
i=1 i i i
P ∈R[t]=R[t ,...,t ]. Let
1 n
U ={x∈A|P(x)(cid:54)=0}, u =#U .
A A A
Then either u =0 or u ≥m(a ,...,a ;a +...+a −degP).
A A 1 n 1 n
When R is a field, this is the Alon-Fu¨redi Theorem [AF93, Thm. 4]. The key
observation that the Combinatorial Nullstellensatz works over an arbitrary ring
provided we impose Condition (D) is due to U. Schauz. It was further developed
in [Cl14, §3]. The relevance of Condition (D) is shown in the following result.
Theorem2.3. (CATS Lemma[Cl14,Thm. 12]) LetRbearing. For1≤i≤n,
let A ⊂ R be nonempty and finite. Put A = (cid:81)r A . For 1 ≤ i ≤ n, let
i i=1 i
(cid:81)
ϕ = (t −a ).
i ai∈Ai i i
a) (Schauz [Sc08]) The following are equivalent:
(i) X satisfies condition (D).
(ii) For all f ∈ R[t ,...,t ], if deg f < #A for all 1 ≤ i ≤ n and f(a) = 0 for
all a∈(cid:81)n A , the1n f =0n. ti i
i=1 i
(iii) If f| ≡0, there are g ,...,g ∈R[t ,...,t ] such that f =(cid:80)n g ϕ .
A 1 n 1 n i=1 i i
b) (Chevalley-Alon-Tarsi) The above conditions hold when R is a domain.
FATTENING UP WARNING’S SECOND THEOREM 9
With Theorem 2.3 in hand, Theorem 2.2 can be established following the original
argumentof[AF93]. However,Ifindthisargumentabitmysterious. Theorem2.2is
the backbone of this work and a key barrier to further generalizations of Theorem
1.7. Because of this I feel the need to give the most conceptually transparent
argumentpossible. ForthisweadaptaproofofAlon-Fu¨rediduetoBallandSerra.
Proof. Step 1: We establish a variant of the Punctured Combinatorial Nullstellen-
satzofBall-Serra[BS09,Thm. 4.1].1 LetRbearing,letA ,...,A ⊂Rsatisfying
1 n
Condition(D),andputA=(cid:81)n A ,Y =(cid:81)n Y . For1≤i≤nlet∅(cid:54)=Y ⊂A .
i=1 i i=1 i i i
For 1≤i≤n, put
(cid:89) (cid:89)
ϕ (t)= (t −a ), ψ (t)= (t −y ).
i i i i i i
ai∈Ai yi∈Yi
Let f ∈ R[t] = R[t ,...,t ]. Suppose that for all x ∈ A\Y, f(x) = 0. Then we
1 n
claim there are g ,...,g ,u∈R[t] such that
1 n
n n n
f =(cid:88)g ϕ +u(cid:89)ϕi, degu≤degf −(cid:88)(#A −#Y ).
i i ψ i i
i
i=1 i=1 i=1
proof of claim: Weperformpolynomialdivisiononf bythemonicpolynomial
ϕ , then divide the remainder by the monic polynomial ϕ , and so forth, finally
1 2
dividing by ϕ to get f = (cid:80)n g ϕ +r. By [Cl14, §3.1], we have degr ≤ degf
n i=1 i i
and deg r <degϕ for all i. Dividing rψ by ϕ we get
ti i 1 1
rψ =r ϕ +s .
1 1 1 1
Then
deg s <degϕ
t1 1 1
whereas for all i(cid:54)=1,
deg s ≤deg rψ =deg r <degϕ .
ti 1 ti 1 ti i
Since s vanishes identically on A and A satisfies Condition (D), Theorem 2.3
1
applies to show s = 0: that is, we may write r = ϕ1r . Continuing this process
with respect to t 1,...,t , we get r =(cid:81)n ϕiu withψ1 1
2 n i=1 ψi
n n
(cid:88) (cid:88)
degu≤degr− (deg(ϕ )−deg(ψ ))≤degf − (#A −#Y ).
i i i i
i=1 i=1
Step 2: Put A = (cid:81)n A , and let f ∈ R[t ,...,t ]. We may assume that f does
i=1 i 1 n
not vanish identically on A. We go by induction on n, the case n = 1 following
from Theorem 2.3. Suppose n≥2 and the result holds for n−1. Define
(cid:40)
A , 1≤i≤n−1
Y = i .
i
{y ∈X |f(t ,...,t ,y)(cid:54)=0} i=n
n 1 n−1
By our hypothesis on f, Y (cid:54)=∅. Let y ∈Y . We apply Step 1 to f, getting
n n
n
f =(cid:88)g ϕ +uϕn
i i ψ
n
i=1
1TheresultestablishedhereisobtainedfromthePuncturedCombinatorialNullstellensatzby
(i)workingoveranarbitraryringunderCondition(D)and(ii)neglectingmultiplicities.
10 PETEL.CLARK
and put w(t ,...,t )=u(t ,...,t ,y). Then
1 n−1 1 n−1
degw ≤degu≤degf −#A +#Y ,
n n
and for all x(cid:48) = (x ,...,x ) ∈ (cid:81)n−1A , we have f(x(cid:48),y) = 0 ⇐⇒ w(x(cid:48)) = 0.
1 n−1 i=1 i
By induction there are a ,...,a ∈Z+ with 1≤a ≤#A for all i and
1 n−1 i i
n−1 (cid:32)n−1 (cid:33) (cid:32)n−1 (cid:33)
(cid:88) (cid:88) (cid:88)
a = #A −degw ≥ #A −degu
i i i
i=1 i=1 i=1
suchthatwisnonvanishingatatleast(cid:81)n−1a pointsof(cid:81)n−1A . Thea ,...,a
i=1 i i=1 i 1 n−1
dependony, butifwechoosea ,...,a soastominimize(cid:81)n−1a , thenwefind
1 n−1 i=1 i
((cid:81)n−1a )(#Y ) points of X at which f is nonvanishing, hence
i=1 i n
n−1
(cid:88)
U ≥m(#A ,...,#A ; a +#Y ).
A 1 n i n
i=1
Since
n−1 n−1 n
(cid:88) (cid:88) (cid:88)
a +#Y ≥( #A )−degu+(degu+#A −degf)= #A −degf,
i n i n i
i=1 i=1 i=1
we have U ≥m(#A ,...,#A ;(cid:80)n #A −degf). (cid:3)
A 1 n i=1 i
Remark 2.4. Theorem 2.2 is sharp in the following sense: let R be a ring,
A ,...,A ⊂ R satisfying Condition (D), and put A = (cid:81)n A . Let d ∈ Z+.
1 n i=1 i
There is a degree d polynomial f ∈ R[t ,...,t ] which is nonzero at precisely
1 n
m(#A ,...,#A ;(cid:80)n #A −d) points of A. In fact something stronger holds: let
1 n i=1 i
y = (y ,...,y ) ∈ Zn with 1 ≤ y ≤ #A for all i. For 1 ≤ i ≤ n, choose Y ⊂ A
1 n i i i i
with #Y =y , and put f =(cid:81)n (cid:81) (t −x). Then degf =(cid:80)n (#A −#Y )
and f isinonvianishing preciselyi=o1n (cid:81)x∈nYi(Ai \Y ), a subset of sizei(cid:81)=1n yi. i
i=1 i i i=1 i
Remark 2.5. Both of the main results of [BS09] – namely Theorems 3.1 and 4.1
– can be generalized by replacing the arbitrary field F by an arbitrary ring R under
theassumptionthatthesetssatisfyCondition(D).Intheformercasetheargument
adapts immediately; in the latter case it requires some mild modifications.
2.3. Proof of the Main Theorem.
Proof. WewillproveTheorem1.7. AsinRemark1.8, wemayassumeR isaDVR,
and thus our assumption on A ,...,A ,B ,...,B becomes Condition (F). Let
1 n 1 r
A=(cid:81)ni=1Ai. For 1≤j ≤r, let Bj be the image of Bj in R/pvj. For a∈Z+, let
S(a) be a set of coset representatives for pa in R. Put
r
(cid:89) (cid:89)
Q(t)= (P (t)−y)∈R[t].
j
j=1y∈S(vj)\Bj
For 1≤j ≤s put
vj−1
(cid:88)
c = (qi−1),
j
i=1
and put
r
(cid:88)
c= c .
j
j=1
