Criteria for Finite Difference Groebner Bases of Normal Binomial Difference Ideals

0.21 MB·

by  Yu-Ao Chen

Criteria for Finite Difference Gröbner Bases of
Normal Binomial Difference Ideals
∗
Yu-AoChen andXiao-Shan Gao
KLMM,UCAS, Academy ofMathematicsand SystemsScience
7 TheChineseAcademy ofSciences, Beijing 100190,China
1
0
2

n
a Abstract
J

3 Inthispaper,we givedecisioncriteriafornormalbinomialdifferencepolynomialidealsintheunivariate
2 difference polynomial ring F y to have finite difference Gröbner bases and an algorithm to compute
the finite difference Gröbner b{as}es if these criteria are satisfied. The novelty of these criteria lies in the
]
C factthatcomplicatedpropertiesaboutdifferencepolynomialidealsarereducedtoelementarypropertiesof
S univariatepolynomialsinZ[x].
.
s Keywords.Differencealgebra,binomialdifferenceideal,Gröbnerbasis,differenceGröbnerbasis.
c
[

1
1 Introduction
v
8
4
DifferencealgebrafoundedbyRittandCohnaimstostudyalgebraicdifferenceequationsinasimilarwaythat
2
6 polynomialequationsarestudiedincommutativealgebraandalgebraicgeometry [5,14,18,21]. TheGröbner
0 basis invented by Buchberger is a powerful tool for solving many mathematical problems [4]. The concepts
.
1 of difference Gröbner bases was extended to linear difference polynomial ideals in [11,14,15] and nonlinear
0
difference polynomialidealsin[11]. Manyapplications ofdifference Gröbnerbasesweregiven[9,14–16].
7
1 Sincedifferencepolynomialidealscanbeinfinitelygenerated,theirdifferenceGröbnerbasesaregenerally
:
v infinite. Even for finitely generated difference polynomial ideals, their difference Gröbner bases could be
i infiniteasshownbyExample2.2inthispaper. Thismakesitimpossible tocomputedifferenceGröbnerbases
X
forgeneraldifference polynomial idealsandthusitisacrucial issuetogivecriteria fordifference polynomial
r
a idealstohavefinitedifference Gröbnerbases.
Let F be a difference field and y a difference indeterminate. In this paper, we will give decision criteria
for normal binomial difference polynomial ideals in F y to have finite difference Gröbner bases and an
{ }
algorithmtocomputethesefinitedifferenceGröbnerbasesunderthesecriteria. AdifferenceidealI inF y
{ }
is called normal if MP I implies P I for any difference monomial M in F y and P F y . I is
∈ ∈ { } ∈ { }
calledbinomialifitisgenerated bydifference polynomials withatmosttwoterms[6,7].
For f Z[x], let f+,f N[x] bethe positive part and the negative part of f such that f = f+ f . For
− −
∈ ∈ −
h=(cid:229) mi=0aixi ∈N[x], denote yh =(cid:213) mi=0(s iy)ai, where s is the difference operator of F. Then any difference
monomial in F y can be written as yg for some g N[x]. For a given f Z[x] with a positive leading
{ } ∈ ∈
coefficient, weconsider thefollowingbinomialdifference polynomial idealinF y :
{ }
If =sat(yf+ yf−)=[ yh+ yh− h=gf,g Z[x] ]
− { − | ∈ }
PartiallysupportedbyagrantfromNSFC11101411.
∗
1
wheresatisthedifference saturation idealtobedefinedinSection2ofthispaper. Let
F , h Z[x] lt(h)=h+ .
0
F , {h∈Z[x]|hg F fo}rsomemonicpolynomial g Z[x] .
1 0
{ ∈ | ∈ ∈ }
WeprovethatI hasafinitedifferenceGröbnerbasisifandonlyif f F . Thiscriterionisthenextendedto
f 1
∈
generalnormalbinomialdifference idealsinF y .
{ }
Thedecision of f F isquitenontrivial andwegivethefollowingcriteriafor f F basedontheroots
1 1
∈ ∈
of f:
1. if f hasnopositiveroots,then f F ;
1
∈
2. if f hasmorethanonepositiveroots(withmultiplicity counted), then f F ;
1
6∈
3. if f hasonepositiverootx andarootzsuchthat z >x ,then f F ;
+ + 1
| | 6∈
4. if f hasonepositiverootx andarootzsuchthat z =x ,thenwecancomputeanother f Z[x]and
+ + ∗
| | ∈
x R suchthat f (x )=0, f (w)=0and w =x implyw=x ,and f (w)=0and w =x imply
∗ >0 ∗ ∗ ∗ ∗ ∗ ∗ ∗
∈ | | | |6
w <x . Furthermore, f F ifandonlyif f F ;
∗ 1 ∗ 1
| | ∈ ∈
5. if f /F hasauniquepositiverealrootx andx <1,then f F ;
0 + + 1
∈ 6∈
6. if f(1)=0andanyotherrootzof f satisfies z <1,then f F ifandonlyif f(x)/(x 1) Z[xd ]for
1
somed N and f(x)(xd 1)/(x 1) F .| | ∈ − ∈
>0 0
∈ − − ∈
With these criteria, only one case is open: f has a unique positive real root x , x >1, and x > z for any
+ + +
| |
other root zof f. Weconjecture that f F in theabove case based onnumerical computations. IfI has a
1 f
∈
finite difference Gröbner basis according to one of the six criteria listed above, we also give an algorithm to
computeit.
Asfarasweknowtheabovecriteriaarethefirstnon-trivial onesforadifference polynomialidealtohave
afinitedifferenceGröbnerbasis. Thenoveltyofthesecriterialiesinthefactthatcomplicatedpropertiesabout
difference polynomialidealsarereducedtoelementary properties ofunivariate polynomials inZ[x].
Therest ofthis paper is organized as follows. In Section 2, preliminaries on Gröbner basis for difference
polynomialidealsaregiven. InSection3,criteriafornormalbinomialdifferenceidealsinF y tohavefinite
{ }
difference Gröbner bases are given. In Section 4, criteria for f F and an algorithm to compute the finite
1
∈
difference Gröbnerbasis ofI under these criteria aregiven. InSection5, wepropose anapproach based on
f
integerprogramming tofindgsuchthat fg F andgivealowerboundfordeg(g)incertain cases.
0
∈
2 Preliminaries on Gröbner basis of difference polynomial ideals
2.1 Gröbner basisofadifference polynomialideal
Anordinary difference field, or simply a s -field, is a field F with a third unitary operation s satisfying: for
anya,b F, s (a+b)=s (a)+s (b), s (ab)=s (a)s (b), and s (a)=0ifandonly ifa=0. Wecalls the
∈
differenceortransformingoperatorofF. Atypicalexampleofs -fieldisQ(l )withs (f(l ))= f(l +1). In
thispaper,weuses -astheabbreviation fordifference ortransformally.
Forainanys -extensionringofF andn N ,s n(a)iscalledthen-thtransformofaanddenotedbyaxn,
>0
∈
withtheusualassumptiona0=1andx0=1. Moregenerally,forp=(cid:229) si=0cixi∈N[x],denoteap=(cid:213) si=0(s ia)ci.
Forinstance, a3x2+x+4=(s 2(a))3s (a)a4. Itiseasytocheckthatap satisfiestheproperties ofpowers[7].
2
LetS beasubset ofas -fieldG whichcontains F. Wewilldenote Q (S)= s kak N,a S , F S =
{ | ∈ ∈ } { }
F[Q (S)]. Now suppose Y= y ,...,y is a set of s -indeterminates over F. The elements of F Y are
1 n
{ } { }
called s -polynomials over F in Y. A s -polynomial ideal I, or simply a s -ideal, in F Y is a possibly
{ }
infinitely generated ordinary algebraic ideal satisfying s (I) I. If Sisasubset of F Y , weuse (S)and
⊂ { }
[S]todenotethealgebraic idealandthes -idealgenerated byS.
A monomial order in F Y is called compatible with the s -structure, if yxk1 <yxk2 for k <k . Only
{ } i j 1 2
compatible monomial orders are considered in this paper. When a monomial order is given, we use LM(P)
and LC(P) to denote the largest monomial and its coefficient in P respectively, and LT(P)=LC(P)LM(P)
theleadingtermofP.
Definition 2.1. G F Y is called a s -Gröbner basis of a s -ideal I if for any P I, there exist m N
andG Gsuchtha⊂t(LM{ (}G))xm LM(P). ∈ ∈
∈ |
From the definition, G is a s -Gröbner basis of I if and only if Q (G) is a Gröbner basis of I treated
as an algebraic polynomial ideal in F[Q (Y)]. Note that I is generally an infinitely generated ideal and the
concept of infinite Gröbner basis [12] is adopted here. From this observation, we may see that a s -Gröbner
basissatisfiesmostofthepropertiesoftheusualalgebraicGröbnerbasis. Forinstance,Gisas -Gröbnerbasis
ofas -idealI ifandonlyifforanyP I,wehavegrem(P,Q (G))=0,wheregrem(P,Q (G))isthenormal
∈
form of P modulo Q (G) in the theory of Gröbner basis. The concepts of reduced s -Gröbner bases could be
similarly introduced. A s -polynomial Q is called s -reduced w.r.t. another s -polynomial P if there does not
existak Nsuch that LM(P)xk divides anymonomial inQ. Then, as -Gröbner Gbasis iscalled reduced, if
anyP G∈iss -reducedw.r.tG P . Itiseasytoseethatas -idealhasauniquereduced s -Grb¨nerbasis.
∈ \{ }
Thefollowingexampleshowsthatevenafinitelygenerated s -idealmayhaveaninfinites -Gröbnerbasis.
Asaconsequence, thereexistnogeneralalgorithmstocomputethes -Gröbnerbasis.
Example 2.2. Let I = [y yx yxy ,y y 1]. Assume y < y < y . Then under a compatible monomial
1 2− 1 2 1 3− 1 2 3
order,thereduced s -GröbnerbasisofI F y ,y is y yxi yxiy i N .
∩ { 1 2} { 1 2 − 1 2| ∈ >0}
2.2 Characteristicsetfora difference polynomialideal
Theelimination ranking R onQ (Y)= s ky 1 i n,k N isused inthis paper: s ky >s ly ifand only
i i j
{ | ≤ ≤ ∈ }
ifi> jori= jandk>l,whichisatotalorderoverQ (Y). Byconvention, 1<s ky forallk N.
j
∈
Let f be a s -polynomial in F Y . The greatest yxk w.r.t. R which appears effectively in f is called
{ } j
the leader of f, denoted by ld(f) and correspondingly y is called the leading variable of f, denoted by
j
lvar(f)=y . The leading coefficient of f as a univariate polynomial in ld(f) is called the initial of f and is
j
denotedbyinit .
f
Let p and q be two s -polynomials in F Y . q is said to be of higher rank than p if ld(q) >ld(p) or
{ }
ld(q)=ld(p)=yxk and deg(q,yxk)>deg(p,yxk). Suppose ld(p)=yxk. qis said to be Ritt-reduced w.r.t. pif
j j j j
deg(q,yxk+l)<deg(p,yxk)foralll N.
j j ∈
A finite sequence of nonzero s -polynomials A :A ,...,A is said to be a difference ascending chain, or
1 m
simplyas -chain,ifm=1andA =0orm>1,A >A andA isRitt-reduced w.r.t. A for1 i< j m. A
1 j i j i
6 ≤ ≤
s -chainA canbewrittenasthefollowingform[8]
A :A ,...,A ,...,A ,...,A (1)
11 1k1 p1 pkp
where lvar(A )=y for j=1,...,k, ord(A ,y )<ord(A ,y ) and deg(A ,ld(A ))>deg(A ,ld(A )) for
ij ci i ij ci il ci ij ij il il
3
j<l. Thefollowingaretwos -chains
A : yx 1, y2y2 1, yx 1
1 1− 1 2− 2− (2)
A : y2 1, yx y , y2 1, yx+y
2 1− 1− 1 2− 2 2
LetA :A ,A ,...,A be as -chain withI asthe initial ofA, and Pany s -polynomial. Then there exists
1 2 t i i
an algorithm, which reduces P w.r.t. A to a s -polynomial R that is Ritt-reduced w.r.t. A and satisfies the
relation
t
(cid:213) Iei P R,mod[A], (3)
i · ≡
i=1
wherethee N[x]andR=prem(P,A)iscalledthes -Ritt-remainderofPw.r.t. A [8].
i
∈
A s -chain C contained in a s -polynomial set S is said to be a characteristic set of S, if S does
not contain any nonzero element Ritt-reduced w.r.t. C. Any s -polynomial set has a characteristic set. A
characteristic setC ofas -idealJ reducestozeroallelementsofJ.
Let A : A ,...,A be a s -chain, I = init(A), yxoi = ld(A). A is called regular if for any j N, Ixj
1 t i i li i ∈ i
is invertible w.r.t A [8] in the sense that [A ,...,A ,Ixj] contains a nonzero s -polynomial involving no
1 i 1 i
yxoi+k,k=0,1,.... Tointroduce the concept of cohere−nt s -chain, weneed to define the D -polynomial first. If
Ali and A have distinct leading variables, wedefine D (A,A )=0. IfA and A (i< j) have the same leading
i j i j i j
variable y ,ld(A)=yxoi,andld(A )=yxoj,theno <o [8]. DefineD (A,A )=prem((A)xoj−oi,A ).ThenA
l i l j l i j i j i j
iscalled coherent ifprem(D (A,A ),A)=0for alli< j [8]. BothA andA in(2)areregular andcoherent
i j 1 2
s -chains.
LetA beas -chain. DenoteI tobetheminimalmultiplicative setcontaining theinitials ofelements of
A
A andtheirtransforms. Thesaturation idealofA isdefinedtobe
sat(A)=[A]:I = P F Y : m I ,mP [A] .
A A
{ ∈ { } ∃ ∈ ∈ }
Thefollowingresultisneededinthispaper.
Theorem2.3. [8,Theorem3.3]As -chainA isacharacteristic setofsat(A)ifandonlyifA isregular and
coherent.
Wealsoneedtheconceptofalgebraicsaturationideal. LetC beanalgebraictriangularsetinF[x ,...,x ]
1 n
andI theproduct oftheinitialsofthepolynomials inC. Thendefine
asat(C)= P F[x ,...,x ] k N,IkP (C) .
1 n
{ ∈ |∃ ∈ ∈ }
2.3 s -Gröbner basisforabinomial s -ideal
A s -monomial in Y can be written as Yf =(cid:213) n yfi, where f=(f ,...,f )t N[x]n. A nonzero vector f=
(f ,...,f )t Z[x]n is said to be normal if thie=1leaiding coefficien1t of f nis p∈ositive, where s is the largest
1 n s
∈
subscriptsuchthat f =0. Forf Z[x]n,letf+,f Nn[x]denoterespectivelythepositivepartandthenegative
s −
partoffsuchthatf=6f+ f−. T∈hengcd(Yf+,Yf−)∈=1foranyf Z[x]n. Iff Z[x]nisnormal,thenYf+ >Yf−
− ∈ ∈
andLT(Yf+ cYf−)=Yf+ underamonomialordercompatible withthes -structure.
−
A s -binomial in Y is a s -polynomial with at most two terms, that is, aYa+bYb where a,b F and
a,b N[x]n. As -idealinF Y iscalledbinomialifitisgeneratedby,possiblyinfinitelymany,s -b∈inomials
∈ { }
[7]. Wehave
4
Proposition 2.4([7]). As -idealI isbinomial ifandonlyifthereduced s -Gröbnerbasis forI consists of
s -binomials.
Let be the multiplicative set generated by yxj for i=1,...,n,j N. A s -ideal I is called normal if
for M m and P F Y , MP I implies P Ii . Normal s -ideals∈in F Y are closely related withthe
∈m ∈ { } ∈ ∈ { }
Z[x]-modulesinZ[x]n [7,13],whichwillbeexplained below. Wefirstintroduce anewconcept.
Definition 2.5. A partial character r on Z[x]n is a homomorphism from a Z[x]-module Lr in Z[x]n to the
multiplicative groupF∗ satisfying r (xf)=(r (f))x =s (r (f))forf Lr .
∈
AZ[x]-module generated byh ,...,h Z[x]n isdenoted as(h ,...,h ) . Letr beapartial character
1 m 1 m Z[x]
∈
over Z[x]n and = f1,...,fs a reduced Gröbner basis of the Z[x]-module Lr =( )Z[x]. For h Z[x]n and
f { } f ∈
H Lr , denote Ph =Yh+ r (h)Yh− and PH = Ph h H . Introduce the following notations associated
⊂ − { | ∈ }
withr :
I+(r ):=[PLr ]=[Yf+−r (f)Yf−|f∈Lr ] (4)
A+(r ):=P = Yf+1 r (f1)Yf−1,...,Yf+s r (fs)Yf−s . (5)
f { − − }
Itisshown that [7]A+(r )is aregular and coherent s -chain and hence is acharacteristic set ofsat(A+(r ))
byTheorem2.3. Furthermore, wehave
Theorem2.6. Thefollowingconditions areequivalent.
1. I isanormalbinomials -idealinF Y .
{ }
2. I =I+(r )forapartial characterr overZ[x]n.
3. I =sat(A+(r ))forapartialcharacterr overZ[x]n.
Furthermore,forf Z[x]n,Yf+ cYf− I f Lr andc=r (f).
∈ − ∈ ⇔ ∈
Asadirectconsequence ofProposition 2.4andTheorem2.6,wehave
Corollary2.7. Letr beapartialcharacter overZ[x]n. ThenP isas -GröbnerbasisofI+(r ).
Lr
Note that for f Z[x]n, either f or f is normal and we need only consider the normal vectors in the s -
∈ −
Gröbner basis. So, for simplicity, we may assume that all given vectors are normal. We have the following
criterionforthes -Gröbnerbasisofnormalbinomials -ideals.
Corollary2.8. Letr beapartialcharacteroverZ[x]n andH Lr . ThenPH isas -GröbnerbasisofI+(r )
⊂
ifandonlyifforanynormalg Lr ,thereexisth H and j N,suchthatg+ xjh+ N[x]n.
∈ ∈ ∈ − ∈
Proof: ByCorollary2.7,P isas -GröbnerbasisofI+(r ). ThenP isas -GröbnerbasisofI+(r )ifand
Lr H
onlyifforanynormal g Lr ,there exist h H and j Nsuch thatLM(xjPh)LM(Pg),whichisequivalent
∈ ∈ ∈ |
tog+ xjh+ N[x]n.
− ∈
Example 2.9. Let f =[1 x,x 1], L= (f) , and r the trivial partial character on L, that is, r (h)=1
Z[x]
− −
for h L. Then P =y yx yxy . By Theorem 2.6, I+(r )=sat(P ). By Corollary 2.7, a s -Gröbner basis
ofI+∈(r )is Yg+f Y1g−2−g=1h2f,h Z[x],lc(h)>0 . ByExample2f.2,sat(Pf)=[Pf,y1y3 1] Q y1,y2 =
[y yxi yxiy {i N− ],an|dareduce∈ds -Gröbnerba}sisofI+(r )is y yxi yxiy i N −. ∩ { }
1 2 − 1 2| ∈ >0 { 1 2 − 1 2| ∈ >0}
5
s
3 Criteria for finite -Gröbner basis
In this section, we will give a criterion for the s -Gröbner basis of a normal binomial s -ideal in F y to be
{ }
finite, where yisas -indeterminate. Without loss ofgenerality, weassume r (h)=1for allpartial characters
r overZ[x]andh Lr .
∈
3.1 Case1: characteristic setcontains a singles -polynomial
Inthissection,weconsiderthesimplestcase: n=1andLr =(f)Z[x] isgeneratedbyonepolynomial f Z[x].
∈
Wewillseethateventhiscaseishighlynontrivial. Forg Z[x],weuselc(g),lm(g),andlt(g)torepresentthe
∈
leadingcoefficient, leadingmonomial,andleading termofg,respectively.
In the rest of this section, we assume f Z[x] and lc(f)>0. Then Pf =yf+ yf− and LT(Pf)=yf+
under a monomial order compatible with the∈s -structure. By Theorem 2.6, all nor−mal binomial s -ideals in
F y whosecharacteristic setconsists ofasingles -polynomialcanbewrittenasthefollowingform:
{ }
If =sat(Pf)=[yh+ yh− h= fg (f)Z[x], (g Z[x],lc(g)>0)]. (6)
− | ∈ ∀ ∈
Inthissection, wewillgiveacriterion forI tohaveafinites -Gröbnerbasis. Define
f
F , f Z[x] lt(f)= f+ .
0
{ ∈ | }
F , f Z[x] fg F forsomemonicpolynomial g Z[x] . (7)
1 0
{ ∈ | ∈ ∈ }
Wenowgivethemainresultofthissection, whichcanbededuced fromLemma3.3andLemma3.7.
Theorem3.1. I in(6)hasafinites -Gröbnerbasisunderamonomialordercompatiblew.r.tthes -structure
f
ifandonlyif f F .
1
∈
For two polynomials h and h Z[x], denote h h if h h N[x]. For h and h N[x], we have
1 2 1 2 1 2 1 2
∈ (cid:23) − ∈ ∈
h1 h2 ifandonlyifyh2 yh1.
(cid:23) |
Lemma3.2. If f F ,then P isas -GröbnerbasisofI .
0 f f
∈ { }
Proof: For g (f) with lc(g) >0, h Z[x] with lc(h) >0 such that g=fh. Since f F , we have
Z[x] 0
∈ ∃ ∈ ∈
lt(f)= f+. Then,
xdeg(h)f+=lt(h)f+/lc(h) lt(h)f+ =lt(h)lt(f)=lt(g) g+.
(cid:22) (cid:22)
ByCorollary2.8, P isas -GröbnerbasisofI .
f f
{ }
Lemma3.3. If f F ,thenI hasafinites -Gröbnerbasis.
1 f
∈
Proof: Let h = fg F , where g is monic. Then lc(h) = lc(f) and lt(h) = lt(f)lm(g) = h+. I =
0 deg(h)
∈
I F[y,yx, ,yxdeg(h)] is apolynomial ideal in a polynomial ring with finitely many variables, which has a
f
T ···
finite Gröbner basis denoted by G6deg(h). Let Pu If and lc(u)>0. If deg(u)6deg(h), then there exists a
∈
P G suchthatt u. Otherwise,wehavedeg(u)>deg(h)andlc(u) lc(f). Then
t 6deg(h)
∈ (cid:22) ≥
xdeg(u) deg(h)h+ = xdeg(u) deg(f) deg(g)lt(f)lm(g)
− − −
= xdeg(u) deg(f)lt(f)=xdeg(u) deg(f)lc(f)lm(f)=lc(f)lm(u) lt(u) u+.
− −
(cid:22) (cid:22)
SincethatPh Ideg(h),byCorollary2.8,G6deg(s) isafinites -GröbnerbasisofIf.
∈
6
Corollary 3.4. Let f F , h=gf F , g a monic polynomial in Z[x] , and D=deg(h). Then the Gröbner
1 0
basisofthepolynomia∈lidealI =∈I F[y,yx, ,yxD]isafinites -GröbnerbasisforI .
D f f
T ···
FromtheproofofLemma3.3,wehave
Example 3.5. f = x2+x+1 F , because (x 1)f = x3 1 F . The finite s -Gröbner basis is G =
1 0
∈ − − ∈
yx2+x+1 1,yx3 y .
{ − − }
LetDbeRorZ. Wewillusethefollowingnewnotation
n
D>0[x], (cid:229) axi n N, i(a D ) .
i i >0
{ | ∈ ∀ ∈ }
i=0
Lemma3.6. N[x] F .
1
⊆
Proof: Let g = a xn+a xn 1+ +a N[x] with d= max d N xd g the multiplicity of f at 0.
n n 1 − 0
Then a >0. Let s=(xn d−+xn d 1·+·· +1∈)g=a x2n d+(a +{a∈ )x|2n |d }1+ +(a + +a )xn+
d − − − n − n n 1 − − n d
(a + +a )xn 1+ +a xd. Rew··r·ite s = b x2n d + +b−xd. Then s/·x·d· Z>0[x·]·.· Let M =
n 1 d − d 2n d − d
ma−x b···/b d+16i·6··2n d +1. Then (x M−)s=b ·x·2·n d+1+(b M∈b )x2n d+ +
i 1 i 2n d − 2n d 1 2n d −
(⌈b {Mb− )x|d+1 Mb xd −F .}S⌉obothsandg−areinF . − − − − − ···
d d+1 d 0 1
− − ∈
Lemma3.7. If f F ,thenI doesnothaveafinites -Gröbnerbasis.
1 f
6∈
Proof: Supposeotherwise,I hasafinites -GröbnerbasisG=P ,whereH= f , ,f Z[x]witheach
f H 1 l
{ ··· }⊂
lc(f)>0. Since f hasthelowestdegreein(f) ,wehave f H.
i Z[x]
∈
Let H , h H lc(h)=lc(f) . Since f / F , we have H F =0/. By Lemmas 3.2 and 3.6, for all
c 1 c 1
{ ∈ | } ∈ T
h H ,h+ hasatleasttwotermsandh hasatleastoneterm. Foru Z[x]withlc(u)>0,defineafunction
c −
∈ ∈
deg(u)=deg(u) (deg(u+ lt(u))) (8)
− −
f
which isthe degree gap between the first twohighest monomials ofu+. Suppose h is an element in H such
1 c
thatdeg(h )=max deg(h) h H . h existsbecause f H =0/ andH isafiniteset. Denotelt(h ),axn,
1 c 1 c c 1
h˜ ,fh lt(h ),lt(h˜{+f),bx|m,∈andh˜˜}+,h˜+ lt(h˜+). Then∈h =6 axn+bxm+h˜˜+ h . Sinceh F ,wehave
1 1− 1 1 1 1 − 1 1 1 − −1 16∈ 1
ab>0. Letc, b/a 1and
⌈ ⌉≥
s=(xn cxm)h =ax2n+xnh˜˜++cxmh (ac b)xm+n cxmh˜+ xnh .
− 1 1 −1 − − − 1 − −1
Wehaves+ s ,ax2n+xnh˜˜++cxmh ,anddeg(s)=deg(s) deg(s+ lt(s)) deg(s )=deg(s ) deg(s+
(cid:22) 0 1 −1 − − ≥ 0 0 − 0−
lt(s ))>n m=deg(h )=deg(h ) deg(h+f lt(h )). f
0 − 1 1 − 1 − 1
SinceP isas f-GröbnerbasisofI ,thereexisth H and j Nsuchthatt=s+ xjh+ N[x]. Weclaim
H f
∈ ∈ − ∈
lt(t)=lt(s+). Ifh H ,thendeg(s)>deg(h). Notethatdeg(s+)=deg(xjh)impliesthatthecoefficientofthe
c
∈
secondlargestmonomialofs+fxjhisnfegativecontradictingtothefacts+ xjh N[x]. Asaconsequence,we
− − ∈
musthave deg(s+)>deg(xjh)and theclaim isproved inthis case. Nowleth H H . Sincelc(h)>lc(s)=
c
∈ \
lc(f), we have deg(xjh)<deg(s) which implies lt(t)=lt(s+). The claim is proved. The fact lt(t)=lt(s+)
implies that when computing the normal form P = grem(P ,Q (P )), we always have lt(u) = lt(s). As a
u s H
consequence, P =0whichcontradicts tothefactthatP isas -GröbnerbasisofI ands (f) .
u H f Z[x]
6 ∈
NotethattheproofofLemma3.7givesamethodtoconstructinfinitelymanyelementsinas -Grb¨nerbasis
asshowninthefollowingexample.
7
Example 3.8. Let f =x2 2x+1 / F . In the proof of Lemma 3.7, c= b/a =1 and s =(x2 1)f =
1 1
− ∈ ⌈ ⌉ −
x4+2x 2x3 1. Repeat the above procedure to s , weobtain s =(x4 2x)s =x8+3x4+2x 2x7 4x2.
1 2 1
− − − − −
Thendeg(f)<deg(s )<deg(s )andP isinas -Gröbnerbasisforalli. Thusanys -GröbnerbasisofI is
1 2 si f
infinitef. Wecanfshowthatfaminimals -Gröbner basisisG= yx2i+1 y2xi i Z yx2i+1+1 yxi+1+xi i
>0
{ − | ∈ }S{ − | ∈
Z .
>0
}
3.2 Finites -Gröbner basesfornormal binomials -ideals
In this section, we consider the general normal binomial s -ideals in F y . By Theorem 2.6, all normal
{ }
binomials -idealsinF y canbewrittenasthefollowingform:
{ }
I =sat(P )=[yg+ yg− g (G) ,lc(g)>0] (9)
G G Z[x]
− |∀ ∈
where
G= g ,...,g Z[x] (10)
1 t
{ }⊂
is a reduced Gröbner basis of the Z[x]-module L=(G) . Gröbner bases in Z[x] have the following special
Z[x]
structure[7].
Lemma 3.9. Let G= g ,...,g be a reduced Gröbner basis of a Z[x]-module in Z[x], g < <g , and
1 k 1 k
{ } ···
lt(gi)=cixdi N[x]. Then
∈
1)0 d <d < <d .
1 2 k
≤ ···
2)c c c andc =c for1 i k 1.
k 2 1 i i+1
|···| | 6 ≤ ≤ −
3) ci g for1 i<k. Ifb istheprimitivepartofg ,thenb g for1<i k.
ck| i ≤ 1 1 1| i ≤
e e
HerearetwoGröbnerbasesinZ[x]: 4,2x , 15,5x,x2+3 .
{ } { }
Intherestofthissection, letL=(G) forGdefinedin(10)anddefine
Z[x]
L , f L lc(f)=c =lc(g ) (11)
i t t
{ ∈ | }
L , f L f hasminimaldegreeinL . (12)
t i i
{ ∈ | }}
Theorem3.10. I hasafinites -GröbnerbasisifandonlyifL F =0/.
G i 0
T 6
Proof: Suppose L F = 0/ and let g L F . Then I k[y,yx, ,yxdeg(g)] has a finite Gröbner basis
i 0 i 0 G
T 6 ∈ T T ···
denoted by G . Let P I and lc(u)>0. If deg(u) deg(g), then there exists a P G such
deg(g) u G h deg(g)
≤ ∈ ≤ ∈ ≤
thath u. Otherwise,wehavedeg(u)>deg(g)andlc(u) lc(g). Then
(cid:22) ≥
xdeg(u) deg(g)g+ =xdeg(u) deg(g)lt(g)=xdeg(u) deg(g)lc(g)lm(g)=lc(g)lm(u) lt(u) u+.
− − −
(cid:22) (cid:22)
ByCorollary2.8,G isafinites -GröbnerbasisofI ,sinceP isinG .
deg(g) G g deg(g)
≤ ≤
We will prove the other direction by contradiction. Suppose that L F = 0/ and I has a finite s -
i 0 G
∩
Gröbnerbasis P = P , ,P . LetH = u , ,u ,andH =H L. Sincegrem(P ,Q (P ))=0,we
H { u1 ··· uk} { 1 ··· k} c T i gt H
have H =0/ and let u be an element of H with maximal deg which is defined in (8). Since L F =0/, by
c 1 c i 0
6 ∩
Lemma3.6 u+ contains at least two terms and u =0. Simfilar to the proof of Lemma 3.7, we can construct
1 −1 6
ans Z[x] Lsuchthatdeg(s)>deg(u )andlc(s)=lc(u ). Then,grem(P ,Q (P ))=0contradicting tothe
1 1 s H
∈ ∩ 6
factthatP isas -Gröbnferbasis. f
H
8
Corollary3.11. IfI hasafinites -Gröbnerbasis,theng F .
G 1 1
∈
Proof: Letb bethe primitive part ofg . ThenbyLemma3.9,b hfor anyh L. ByTheorem 3.10, b and
1 1 1 1
| ∈
henceg isienF . e e
1 1
Corollary3.12. IfL F =0/ andinparticular g F ,thenI hasfinites -GröbnerBasis.
t 1 t 1 G
T 6 ∈
Thefollowingexampleshowsthatg F isnotanecessaryconditionforthes -Gröbnerbasistobefinite.
t 1
∈
Example3.13. LetG= 2(x2 2),(x2 2)(x+1) . Then(x2 2)(x+1)(x 1)+2(x2 2)=x4 x2 2
{ − − } − − − − − ∈
F F ,andhence I has afinite s -Gröbner basis. Ontheother hand, wewillshow (x2 2)(x+1) / F
0 1 G 1
⊂ − ∈
inExample4.10.
Inordertogiveanothercriterion, weneedthefollowingeffectivePolyaTheorem.
n
Lemma3.14([17]). Supposethat f(x)= (cid:229) a xn R[x]ispositiveon[0,¥ )andF(x,y)thehomogenization
n
∈
j=0
of f. Then for Nf > n(n2−l1)L −n, (1+x)Nf f(x) ∈ R>0[x], where l = min{F(x,1−x)|x ∈ [0,1]} and L =
max{k!(nn−!k)!|ak|}.
Corollary3.15. Ifthereexistsanh Lwithnopositiverealroots,thenI hasafinites -Gröbnerbasis.
G
∈
Proof: Writeh=xm1h1suchthath1(0)=0. ByLemma3.14,thereexistsanN Nsuchthath2=(x+1)Nh
6 ∈ ∈
Z>0[x]. Take a sufficiently large N such that deg(h )>d =deg(g ). Then there exists a sufficiently large
2 t t
M N, such that g=xm1(xdeg(h2)−deg(gt)+1gt Mh2) F 0. Since g Li, by Lemma 3.10, I has a finite
∈ − ∈ ∈
s -GröbnerBasis.
F s
4 Membership decision for and -Gröbner basis computation
1
InSection 3,weprovethatsat(P )hasafinites -Gröbner basis ifandonlyif f F . Inthissection, wewill
f 1
∈
givecriteriaandanalgorithmfor f F . If f F ,wealsogiveanalgorithmtocomputethefinites -Gröbner
1 1
∈ ∈
basis.
From the definition of F , a necessarily condition for f F is lc(f)>0. Also, it is easy to show that
1 1
∈
f F ifandonlyifcxmf F forpositiveintegerscandm. Sointherestofthispaper, weassume
1 1
∈ ∈
n
f = (cid:229) a xi Z[x]
n
∈
i=0
suchthatn>0,lc(f)=a >0, f(0)=a =0,andgcd(a ,a ,...,a )=1.
n 0 0 1 n
6
4.1 Decisioncriteria
Inthissubsection, wewillstudywhether f F byexaminingproperties oftherootsof f(x)=0.
1
∈
Lemma4.1. If f Z[x]hasnopositiverealroots,then f F .
1
∈ ∈
Proof: By Lemma 3.14, there exists an N N, such that (x+1)Nf Z>0[x] N[x]. By Lemma 3.6, (x+
∈ ∈ ⊆
1)Nf N[x] F ,andthus f F .
1 1
∈ ⊆ ∈
ByLemma4.1,weneedonlyconsiderthosepolynomials whichhavepositiveroots.
9
Lemma4.2. Let f =a xn+ +a F . Then f hasasimpleandunique positive realrootx ,andforany
n 0 0 +
··· ∈
rootzof f,wehave z x .
+
| |≤
Proof: Since f F Z, the number of sign differences of f is one. Then by Descartes’ rule of signs [1],
0
∈ \
thenumber ofpositive realroots of f (withmultiplicities counted) isoneorlessthanonebyanevennumber.
Then f has a simple and unique positive real root x . Forany root z of f, since a 0 for i=0,...,n 1,
+ i
− ≥ −
wehave
a zn= a zn = a zn 1 a a zn 1 a . (13)
n n n 1 − 0 n 1 − 0
| | | | |− − −···− |≤− − | | −···−
Thus f(z)60andhence f hasatleastonerealrootin[z,¥ ). Since f hasauniquepositiverealrootx ,we
+
| | | |
have z 6x .
+
| |
Wenowconsider those f whichhasarootz=x and z =x . Suchazmustbeeither x oracomplex
+ + +
6 | | −
root.
Lemma 4.3. Let f =a xn+ +a F and x the unique positive root of f. If f has a root z=x but
n 0 0 + +
··· ∈ 6
z =x ,thenwehave
+
| |
1. zd f R>0 andzisasimplerootof f,whered f =gcd i ai =0 >1.
∈ { | 6 }
d d
2. f is apolynomial in x f: f = f x f, where isthe function composition. Furthermore, f(w)=0and
d d ◦ ◦
w =x f implyw=x f. b b
+ +
| |
2pi
3. f hasexactlyd f rootswithabsolute valuex+: z f(z)=0, z =x+ = z kx+ z =edf ,k=1,...,d f ,
{ | | | } { | }
wherei=√ 1.
−
Proof: Letz=x bearootof f suchthat z =x . Then f(z)= f(x )=a zn+a zn 1+ +a =0,
+ + + n n 1 − 0
which, combi6ning with(13),implies a | |zn 1 a |=| a zn 1 | | a .−T|he| above··e·quation is
n 1 − 0 n 1 − 0
possibleifandonlyif azi R for|e−ach−i n −1·a·n·d−a =| 0.−Also−n|o|te,zn−=··(·−a zn 1 a )/a
i >0 i n 1 − 0 n
R . Then, zi R −for ea∈ch i n and a =≤0.−Note that6 zm R and zk R− i−mp|l|y zm−k···−R . As∈a
>0 >0 i >0 >0 − >0
consequence, z∈d f R>0 ford f =≤gcd i ai =6 0 . Sincez=x+,w∈ehaved f >1∈. Part1ofthelemm∈aisproved.
∈ { | 6 } 6
From the definition of d f, f is a polynomial of xd f: f(x)= f(x) (xd f). It is easy to see that f(x) F 0.
◦ d ∈
Let f(x)=b xk+ +b x+b . Thengcd j b =0 =1. Bythbefirstpartofthislemma,weknowbx f isthe
onlybroot of kf who·s·e· abs1olute0value is xd f{. S|injce6 zd }f and xd f are both the unique positive real roots o+f f(x),
+ +
wehavezd f =xd f andhencezisasimplerootof f. Part2ofthelemmaisproved. Part3ofthelemmacobmes
+
fromthefactzd f =xd f istheuniquepositiverealrootof f and f(z)= f(zd f)=0.
+
b
Corollary 4.4. If f F has at least one positive real root x , then x is the unique positive real root of f,
1 + +
∈
x is simple and for any root z of f, x > z. If f has a root z=x satisfying z =x , then z is simple, and
+ + + +
zd R forsomed N ,orequivalentl|y,|theargumentofzs6atisfies Arg(z)/p| | Q.
>0 >1
∈ ∈ ∈
Example4.5. f =(x2 5)(x2 2x+5) / F ,because therootz=1+2isatisfies z =√5butzd / R for
1 >0
− − ∈ | | ∈
anyd N.
∈
Thefollowingexampleshowsthatthemultiplicity forarootzsatisfying z <x couldbeanynumber.
+
| |
Example4.6. Forany n,k N ,(x+1)n(x k) F . Letn=1, (x+1)(x k) F . Let f (x)=(x+1)2
>1 1 0 1
∈ − ∈ − ∈
and fn+1(x) = fn(x)(x2⌊deg(fn)/2⌋+1+1) for n>1. Then we have (x+1)n+1 fn(x), fn(x) Z>0[x], and all
| ∈
coefficients of f areeither1or2. Thus, f (x)(x k) F and(x+1)n(x k) F bydefinition.
n n 0 1
− ∈ − ∈
10