Bézout theorem for a graded ideal in a ring of generalized polynomials
aa r X i v : . [ m a t h . A C ] J u l B ´EZOUT THEOREMFOR A GRADED IDEAL IN A RING OFGENERALIZED POLYNOMIALS.
M. V. KONDRATIEVA
Abstract.
The article proved the upper bound of leading co-efficient of characteristic polynomial of graded ideal in a ring ofgeneralized polynomials.Examples of such rings are as well the rings of commutative poly-nomials (for which the classical B´ezout theorem holds), as somerings of differential operators. For a system of generalized homoge-neous equations in small codimensions we obtain exact polynomialin d estimates. In the general case, the estimate is double expo-nential in τ : O ( d τ − ) , where d is a maximal degree of generatorsof a graded ideal, τ is it’s codimension.For systems of linear differential equations bounds of the sameasymptotics, but by other methods, were obtained by D.Grigoryevin [2].Keywords: Differential algebra, ring of generalized polynomials,graded ideal, characteristic polynomial, typical dimension, B´ezouttheorem. Introduction
In algebraic geometry and commutative algebra many studies aredevoted to the Hilbert polynomial. Differential dimension polynomialwas introduced by E. Kolchin [4] and has the same important rolein differential algebra. Estimations of its coefficients are the classicunsolved problems of differential algebra.In recent years interest in computer algebra has increased, and oneof its directions is the study of Gr¨obner bases. For polynomial ideals,this notion was studied quite fully, in particular upper and lower degreebounds of polynomials in the Gr¨obner basis by degrees of generatorsof the ideal (see, for example, [1]) are found. It is interesting, thatthe complexity of Gr¨obner basis computation and of task of findingthe leading coefficient of Hilbert polynomial has different asymptoticorder.For rings of differential operators over a field, great success in suchstudies were reached by D.Grigoryev and A. Chistov (see [2], [3]). Herethe situation is another, than for polynomial ideals, and is known onlyupper double exponential bound (as for Gr¨obner basis orders, and forthe leading coefficient of dimension polynomial).
M. V. KONDRATIEVA
To generalize these results, we will consider rings, introduced in [5],see definition 4.1.4. Gr¨obner basis technique works for ideals of such(in the general case, non-commutative) rings, and the concept of char-acteristic polynomial is defined.Our bound (Theorem 3) in a situation when degree of characteristicpolynomial 1 less than the maximum possible (we will use the termcodimension 1) coincides with Kolchin⥪s result (see [4], p. 199).By analogy with this linear estimate, Kolchin believed that in othercodimension τ the bound of the leading coefficient also is a polynomialof degree O ( τ ) . In the general case, this has not yet been disproved.In codimensions 2 for a differential dimensional polynomial thebound is proved (see 5.6.7, [5]), and it coincides with that obtainedin this paper for homogeneous systems generalized polynomials.Now in codimensions 3, 4, and 5 (see Theorem 4) exact upper boundsare obtained for the first time. Note (see the example 6) that in codi-mension 3 the bound is achieved.In the case when the characteristic polynomial of a homogeneousideal in the ring of generalized polynomials is a constant, we obtainthe upper double exponential bound of it by the number of generalizedunknowns. This result is similar to the result of D. Grigoriev.2. Preliminary facts.
One can find basic concepts and facts in [4, 9, 5].Denote the set of integers by Z , non-negative integers by N andbinomial coefficients s ( s − ... ( s − m +1) m ! by (cid:0) sm (cid:1) .For vector e = ( j , . . . , j m ) ∈ N m , the order of e is defined by ord e = P mk =1 j k . Note that any numerical polynomial v ( s ) can be writtenas v ( s ) = P di a i (cid:0) s + ii (cid:1) , where a i ∈ Z . We call numbers ( a d , . . . , a ) standard coefficients of polynomial v ( s ) . Definition 1. (see[5], definition 2.4.9 or [6]). Let ω = ω ( s ) be aunivariate numerical polynomial in s and let d = deg ω . The sequenceof minimizing coefficients b ( ω ) is the vector b ( ω ) = ( b d , . . . , b ) ∈ Z d +1 defined by induction on d as follows. If d = 0 (i.e. ω is a constant),then b ( ω ) = ( ω ) . Let d > and ω ( s ) = P di =0 a i (cid:0) s + ii (cid:1) . Let v ( s ) = ω ( s + a d ) − (cid:0) s +1+ d + a d d +1 (cid:1) + (cid:0) s + d +1 d +1 (cid:1) . Since deg v < d , one may suppose thatthe sequence of minimizing coefficients b ( v ) = ( b k , . . . , b ) (0 ≤ k < d ) of the polynomial v ( s ) has been defined. To define the same for ω weset b ( ω ) = ( a d , , . . . , , b k , . . . , b ) ∈ Z d +1 .Now we define the Kolchin dimension polynomial of a subset E ⊂ N m . Definition 2.
Regard the following partial order on N m : the relation ( i , . . . , i m ) ( j , . . . , j m ) is equivalent to i k j k for all k = 1 , . . . , m .We consider a function ω E ( s ) , that in a point s equals Card V E ( s ) , where V E ( s ) is the set of points x ∈ N m such that ord x s and forevery e ∈ E the condition e x isn’t true. Then (see for example, [4],p.115, or [5], theorem 5.4.1) function ω E ( s ) for all sufficiently large s is anumerical polynomial. We call this polynomial the Kolchin dimensionpolynomial of a subset E .Not every numerical polynomial is a Kolchin dimensional polynomialfor some set E . The connection of these concepts is established in thefollowing theorem. Theorem 1. (see [6] and [5] , proposition 2.4.10). The sequence of min-imizing coefficients of dimension polynomial Kolchin consists of onlynon-negative integers. The converse is also true: if the sequence ofminimizing coefficients of some numerical polynomial consists of non-negative numbers, then it is the Kolchin dimension polynomial of someset E . We denote the set of such polynomials by W . Note that the set W is closed with respect to addition, difference ∆ ω ( s ) = ω ( s ) − ω ( s − (1)and positive shift: ( ω ( s ) = ω ( s + j ) , j ∈ N ). (see[5], propositions2.4.13 Рҷ 2.4.22).Let X = { x , . . . , x m } be a finite system of elements. By T = T ( X ) we denote the free commutative semigroup with unity (written mul-tiplicatively), generated by the elements of X . Elements of T will becalled monomials . Let θ ∈ T , θ = x e . . . x e m m . By the order of θ weshall call the sum e + · · · + e m that will be denoted by ord θ . Suppose,that the set of monomials is linearly ordered and for any θ ∈ T thefollowing conditions hold: ≤ θ ; and if θ < θ , then θθ < θθ . In this case we shall say, that a ranking is defined on the set of mono-mials T .Let F be a field, P the vector F -space with the basis T = T ( X ) .We define on P the function “ taking the leader ” in the followingway: any g in P may be represented as a sum g = P θ ∈ T a θ θ , whereonly a finite number of coefficients a θ ∈ F are distinct from zero (suchrepresentation is unique up to the order of the terms). Among all mono-mials, present in this expression with nonzero coefficients, we choosethe maximal with respect to the order introduced on the set T . Thismonomial will be called the leader of g ∈ P and will be denoted by u g . Definition 3.
Let some ranking on the set of monomials T = T ( X ) begiven and let P be the vector F -space with the basis T . Suppose that P is a F -algebra, and u AB = u A u B for all A, B ∈ P . Furthermore, M. V. KONDRATIEVA suppose that θ · θ = 1 θ θ ∈ P for any θ , θ ∈ T ; in particular,the generators x , . . . , x m pairwise commute. Such ring we shall callthe ring of generalized polynomials in the indeterminates X = { x , . . . , x m } .Example 1. The ring of commutative polynomials over a field.
Con-sider an arbitrary ranking on the set X = { x , . . . , x m } . Let P bethe algebra F [ x , . . . , x m ] of polynomials in the commutative indeter-minates x , . . . , x m over a field F . It is easy to see, that the condi-tion u AB = u A u B holds for all A, B ∈ P and therefore we may treat F [ x , . . . , x m ] as a ring of generalized polynomials in the indeterminates x , . . . , x m . Definition 4.
An operator ∂ on a ring K is called a derivation op-erator (or differentiation ) iff ∂ ( a + b ) = ∂ ( a ) + ∂ ( b ) and ∂ ( ab ) = a∂ ( b ) + ∂ ( a ) b for all a, b ∈ K .A commutative ring K with a finite set ∆ = { ∂ , . . . , ∂ m } of mutuallycommuting derivation operators on K is called a differential ring . Definition 5.
Let F be a differential field and let ∆ = { ∂ , . . . , ∂ m } bea basic set of derivation operators on F . The ring D = F [ ∂ , . . . , ∂ m ] of skew polynomials in indeterminates ∂ , . . . , ∂ m with coefficients in F and the commutation rules ∂ i ∂ j = ∂ j ∂ i , ∂ i a = a∂ i + ∂ i ( a ) for all a ∈ F , ∂ i , ∂ j ∈ ∆ is called a (linear) differential (or ∆ -) operatorring . on F . Example 2.
The ring of differential operators over a field.
Let F be a ∆ -field, and let an arbitrary ranking be fixed on the set T = T (∆) . Then the ring D = F [ ∂ , . . . , ∂ m ] of linear differential operatorsover F (see definition 5) is a ring of generalized polynomials in theindeterminates ∂ , . . . , ∂ m . Example 3.
The ring of differential operators over a ring of polyno-mials.
Let F be a ∆ = { ∂ , . . . ∂ m } -field and R a ring of commutativepolynomials in the indeterminates y , . . . , y n over F . We define thederivation operators ∆ ′ = { ∂ ′ , . . . ∂ ′ m } on R in the following way. Set ∂ ′ i ( f ) = 0 for all i = 1 , . . . , m , j = 1 , . . . , n . For any i m wefix a number j n (for different i the corresponding indices j may coincide) and set ∂ ′ i ( f ) = ∂ i ( f ) y j for all j = 1 , . . . , n and f ∈ F .Then the ring D R of linear ∆ ′ -operators over R is a ring of generalizedpolynomials in the indeterminates X = { ∂ ′ , . . . , ∂ ′ m , y , . . . , y n } . In-deed, if we consider the ranking such that ∂ ′ i > y j for all i = 1 , . . . , m , j = 1 , . . . , n , then the condition u f u g = u fg is fulfilled.Let D be a ring of generalized polynomials in the indeterminates X = { x , . . . , x m } over a field F and F the free D -module with thebasis B = { f , . . . , f n } . The F -vector space F has as a basis the direct(Cartesian) product T × B of the sets T = T ( X ) and B . This product we shall call the set of terms of the module F , T F = { x i . . . x i m m f j | ( i , . . . , i m ) ∈ N m , j = 1 , . . . , n } . We cannot multiply terms, but we can define the product of a term bya monomial satisfying the following conditions:for any term u ≤ v, u, v ∈ T F , and for any monomial θ ∈ T true θu ≤ θv. Definition 6.
A ranking will be called orderly if the condition ord θ < ord θ ( θ , θ ∈ T ) implies θ f i < θ f j for all ≤ i, j ≤ n . Example 4.
Let a ranking on the set T of monomials be given. Weshall order the terms T F : θ f i < θ f j if either i < j or i = j and θ < θ . Such ranking on T F is not orderly. Example 5.
Let a ranking on the set T is following: θ < θ iffeither ord θ < ord θ , or ord θ = ord θ and θ < θ with re-spect to lexicographic order on monomials. Let t , t ∈ T F . Weset t = θ f i < t = θ f j if and only if either ord θ < ord θ , or ord θ = ord θ and i < j , or ord θ = ord θ , i = j and θ < θ . Thisranking is orderly. We shall call it standard .In the submodule of the free module F over the ring of generalizedpolynomials D a Gr¨obner basis exists: Definition 7. (see definition 4.1.25, [5]). Let D be a ring of generalizedpolynomials in indeterminates X = { x , . . . , x m } , F a free D -module.Suppose that M ⊆ F is a submodule of F , G ⊂ M is a finite set and < is a ranking on the set of terms T F . The set G is called a Gr¨obnerbasis of M , if there exists for any nonzero f ∈ M a representation: f = r X i =1 c i θ i g i , = c i ∈ F , θ i ∈ T ( X ) , g i ∈ G, θ i u g i > θ i +1 u g i +1 , that, in particular, implies u f = θ u g . We shall now consider graded modules over the ring of generalizedpolynomials. Firstly we consider in T = T ( X ) the subset T s = ( x i . . . x i m m | m X k =1 i k = s, ( i , . . . , i m ) ∈ N m ) ,s ∈ Z and T s = for all s < . Definition 8.
Let D be a ring of generalized polynomials over a field K in the indeterminates X = { x , . . . , x m } We suppose the ranking of T = T ( X ) to be orderly, and D s = nX θ ∈ T s a θ θ | a θ ∈ F and almost all coefficients are equal to 0. o M. V. KONDRATIEVA
The ring D will be called graded if D = M s ∈ N D s and D s D r ⊆ D s + r for all s, r ∈ N .The rings (examples 1, 4), with standart ranking (see example 5),are graded. The ring of differential operators over field F (example 6)isn’t graded, if there are non-constant elements in field F . Definition 9.
Let D be a graded ring of generalized polynomials overa field F . A D -module M will be called graded , if for any s ∈ N a F -subspace M s of M is defined such M = L s ∈ N M s and D s M r ⊆ M s + r for all s, r ∈ N . The elements of M s will be called the homoge-neous elements of degree s . Definition 10.
Let D M be a finitely generated module over a ring ofgeneralized polynomials and M = L s ∈ N M s be a grading of M . Thefunction φ grM , whose value at any s ∈ N is equal to dim F M s will becalled the characteristic function of the graded module M . Theorem 2. (see [5] , theorem 4.3.20.) Let D be a graded ring of gener-alized polynomials over a field in the indeterminates X = { x , . . . , x m } , D M be a graded module and { m , . . . , m n } be a finite set of its genera-tors such that m i ∈ M ( α ) i . Then there exist sets E i ⊂ N m ( i = 1 , . . . , n ) such that for all large enough s the characteristic function of M is equalto φ grM ( s ) = ∆ n X i =1 ω E i ( s − α i ) , (2) where ω E ( s ) is the Kolchin dimension polynomial of the matrix E (seetheorem 2, equation 1). As follows from the proof, the sets E i correspond to leaders of ahomogeneous Gr¨obner basis of relations between generators (syzygymodule). It is easy to see that for sufficiently large s the function φ grM ( s ) is polynomial.We denote it by ω M ( s ) and call the characteristic polynomial graded finitely generated module D M . Its degree d ( M ) = deg( ω M ) is called (generalized) type of module M , the difference ( m − − d ) - (generalized) codimension , and the standard leading (nonzero)coefficient τ d ( M ) - (generalized) type dimension. Graded modules over a ring of generalized polynomials have prop-erties similar to properties differential modules: d ( M ) < m and a m − ( ω M ) = rk D M . Let F be a free D -module with generators f , . . . , f n . Each elementof f ∈ F is represented as f = P j n θ j f j , where θ j ∈ D . Denote by ord f i f = ord θ i and ord f = max i n (ord θ i ) .Consider the following grading on F : F s = P ni =1 T s f i . Let H bethe submodule of the module F generated by elements of Σ ⊂ H , and ord f j h e j for all j = 1 , . . . , n, h ∈ Σ . The induced grading ariseson the module H : H s = H ∩ F s . The factor module F/H can alsobe regarded as graded: ( F/H ) s = ( F s /H s ) , and ω F/H ( s ) = ω F ( s ) − ω H ( s ) = P ni =1 (cid:0) s + m − m − (cid:1) − ω M ( s ) . Sometimes this polynomial calledthe characteristic polynomial of a system of generalized polynomialequations Σ (or a system of D -equations) and denoted by ω [Σ] .As follows from the theorem 2 (put a i = 0 ( i = 1 , . . . , n ) , M = F/H ), characteristic polynomial of a system of generalized polynomialequations can be calculated as in the differential case: (Theorem 4.3.5[5]): ω [Σ] ( s ) = n X j =1 ∆ ω E j ( s ) , (3)where E j ⊂ N m .We are interested in following Question 1.
How to estimate the typical dimension of Σ in knownorders e , . . . , e n ?Firstly this question was asked in differential algebra by J.Ritt forordinary differential systems. Later E.Kolchin decided this problem ina codimension for nonlinear systems. His bound (see [4], p. 199) is asfollows: typical differential dimension a m − of the system Σ does notexceed e + · · · + e n .In codimension 2, such a result is known (see 5.6.7, [5]):Let n = 1 , then a m − ( ω Σ ) e .Both of these results are also true for systems of homogeneous gen-eralized polynomial equations.3. Basic results.
So, for systems of generalized homogeneous polynomial equationsin codimensions 1 and 2, the classical B´ezout theorem holds. If thecodimension is greater than 2, in the general case this is not true.Consider an example.
Example 6.
Let k ∈ N , F = C ( x , x , x ) , n = 1 , D = F { ∂ , ∂ , ∂ , y } –ring of differential operators over ring of polynomials in one variable y , (see the example 4) over the field F , m = 4 , Σ = { ∂ k f , ( ∂ k + x ∂ k ) f } . M. V. KONDRATIEVA
Let’s we have the orderly rank ∂ > ∂ > ∂ > y . One can find ahomogeneous Gr¨obner basis of the ideal [Σ] . It consists of elements G = { ∂ k f , ( ∂ k + x ∂ k ) f , ∂ k − ∂ k y f , ∂ k − ∂ k y f , ..., ∂ k − i ∂ i ∗ k y i f ,..., ∂ k y k f } ,From here according to the equation (3), ω [Σ] = ∆ ω E , where E = k k k − k . . . . . . . . . . . .k − i ik i. . . . . . . . . . . . k k . One of the main methods for calculating the dimension polynomialof a matrix is the using of the formula (see [5], Theorem 2.2.10): ω E ( s ) = ω E ∪ e + ω H ( s − ord( e )) , (4)where e ∈ N m , H is the matrix obtained by subtracting the vector e from each row of E (negative numbers are replaced by zeros).Apply the formula (4) e = (0 , k, , k times, we obtain ω E = kω E ,here E = k k − k . . . . . . . . .k − i ik i. . . . . . . . . k k . By Theorem 2.2.17 (see [5]) we have: ∆ ( ω E ) = ω E + ω E , where E , E are the matrixs, obtained bydeletion, respectively, second and third columns of the matrix E . Ap-plying corollary 2.3.21 (see [5]), we get ω E = 1+2+ · · · + k = k ( k +1) / and ω E = k (1 + 2 + · · · + k ) = k ( k + 1) / , whence ω E = k ( k + 1) / .If the classical B´ezout theorem holds for the system Σ , we would haveto have ω [Σ] = k ( k + 1) / ≤ k (the system has codimension 3, whileit has 2 homogeneous generators), which is wrong. Theorem 3.
Let D be a graded ring of generalized polynomials overthe field F in indeterminates X = { x , . . . , x m } , F = L ni =1 D - freegraded D -module with generators f , . . . , f n , Σ ⊂ F is a system ofhomogeneous D -equations. Let be ord f j h e j for any h ∈ Σ .Then the following statements are true:if the codimension of the system is 0, then typical differential dimen-sion does not exceed n ;if the codimension of the system Σ is 1, then τ d (Σ) e + dots + e n ;if the codimension of d (Σ) is 2, then τ d (Σ) ( e + · · · + e n ) max ni =1 e i + prod i Lemma 1. Suppose that under the conditions of the theorem 3 thegeneralized type of the system Σ is greater than 1. Then ω Σ ( s ) = n X i =1 (cid:18)(cid:18) s + m − m − (cid:19) − (cid:18) s + m − − ˜ e i m − (cid:19)(cid:19) − w ( s ) , (5) where w ( s + e ) ∈ W , e = max ni =1 ( e i ) , ˜ e i e i .Proof. We denote by H the submodule of the D -module F generatedby the system Σ . D is an Ore ring and, since the codimension of Σ is greater than 1, rk D F/H = 0 , whence rk D H = n . We choose nD -independent equations from Σ , and let D M - D -factor module F bythese equations (we denote them by Σ . We have the exact sequenceof graded D -modules: → N → M − > → F/H → . (6)We can assume that N is a graded submodule of the module M ,generated by the equations Σ \ Σ , and let α i - the degrees of thesegenerators. Gradings, associated with the choice of homogeneous gen-erators ( N s = D s − α i g i ) and grading, induced by M ( N s = M s ∩ N )coincide.Let e = max( e , . . . , e n ) . From the theorem 2 it follows that ω N ( s + e ) ∈ W . Indeed, the generators g i of the module N have degrees α i not greater than e i and from the formula (2) we get that ω N ( s + e ) = P ki =1 ω i ( s + e − α i ) , where ω i ∈ W (here k = Card(Σ) − n ) .Because α i e i , keeping in mind closed W relatively positive shift andsummation we get that ω N ( s + e ) ∈ W .To calculate ω M = ω Σ we use D -independence of the equations Σ .Proof of Lemma 5.8.2 ([5]) can also be done for a system of generalizedpolynomial equations, therefore we have: J (Σ ) = ∞ , where J is theJacobi number of matrix (ord f i h j ) ni,j =1 , h i ∈ Σ . Choosing the finaldiagonal sum of the matrix and renumbering the equations Σ , since ord h f i h i = ∞ , we get ω Σ ( s ) = ω F ( s ) − n X i =1 (cid:18) s + m − − ord h i m − (cid:19) = n X i =1 (cid:18) s + m − m − (cid:19) − n X i =1 (cid:18) s + m − − ˜ e i m − (cid:19) , where ˜ e i = ord f i h i e i .To prove the lemma it remains to use the equality ω M = ω N + ω F/H ,obtained from the sequence (6). (cid:3) We return to the proof of the theorem. Proof. The case d (Σ) = m − follows from Theorems 2.Let d (Σ) = m − . It follows from the lemma 5 that P ni =1 (( s +1) − ( s +1 − ˜ e i )) − ∆ m − ω Σ ( s ) ∈ W (since W is closed relative to the operation ∆ , see the formula (1)). From here P ni =1 ˜ e i − τ d (Σ) ∈ W . Becauseminimizing coefficients of a polynomial from W are non-negative, weimmediately get that τ d (Σ) P ni =1 ˜ e i P ni =1 e i .Let d (Σ) = m − . As above, we use the operator ∆ m − to expres-sion (5). We get: ∆ m − n X i =1 (cid:18) s + m − m − (cid:19) − (cid:18) s + m − − ˜ e i m − (cid:19)! − τ d (Σ) = w ′ ( s ) ,w ′ ( s + e ) ∈ W, whence n X i =1 (cid:18) s + 22 (cid:19) − (cid:18) s + 2 − ˜ e i (cid:19)! − τ d (Σ) = w ′ ( s ) , w ′ ( s + e ) ∈ W and P ni =1 (˜ e i ( s + 1) − (cid:0) ˜ e i (cid:1) ) − τ d (Σ) = w ′ ( s ) . Let the minimizingcoefficients of the polynomial w ′ ( s + e ) be equal to ( b , b ) . Then w ′ ( s + e ) = b ( s + 1) − (cid:0) b (cid:1) + b , and b > = 0 , b > = 0 . We have: τ d (Σ) = ( P ni =1 ˜ e i − b )( s + 1) − P ni =1 (˜ e i − (cid:0) ˜ e i (cid:1) ) + eb + (cid:0) b (cid:1) − b .Equating the coefficient in s to the right side of the equality to zero,we obtain b = P ni =1 ˜ e i , whence τ d (Σ) ( P ni =1 ˜ e i ) e − P ni =1 (cid:0) ˜ e i (cid:1) + (cid:0) P ni =1 ˜ e i (cid:1) = Q i Let D be a graded ring of generalized polynomials overthe field F in indeterminates X = { x , . . . , x m } , Σ ⊂ D is a system ofhomogeneous D -equations. Let be ord h e for any h ∈ Σ .Then the following bounds are true:if the codimension of Σ is 3, then τ d (Σ) e ( e + 1) / (according tothe example 6 this estimate is achieved);if the codimension of Σ is 4, then τ d (Σ) e ( e + 1) (3 e + 6 e +11 e + 8 e + 8) / ;if the codimension of Σ is 5, then τ d (Σ) e ( e + 1) (288 + 480 e +952 e + 1264 e + 1592 e + 1648 e + 1529 e + 1174 e + 775 e + 420 e +183 e e e e + 1) / in any codimension τ > the generalized typical dimension τ d (Σ) does not exceed O ( e τ − ) .Proof. is based on the lemma 1 and the fact that minimizing coefficientsare multivalued from the set W are non-negative. For Σ ′ , we choosethe element Σ in maximal order, let it be e . Then, in the equation (5) n = 1 , ˜ e = e .Consider the case of codimension 3. We apply the operator ∆ m − toboth sides of the equality (5). τ d (Σ) = (cid:18)(cid:18) s + 33 (cid:19) − (cid:18) s + 3 − e (cid:19)(cid:19) − w ′ ( s ) , w ′ ( s + e ) ∈ W Let the sequence of minimizing coefficients of the polynomial w ′ is ( b , b , b ) . According to the definition 1, we can explicitly expressthe standard coefficients w ′ through the numbers b , b , b and findthe coefficients of ’shifted’ the polynomial w ′ ( s + e ) . Equating thecoefficients at s , s on the right side of the equation to 0, we get: b = e , b = e and τ d (Σ) = (cid:18)(cid:18) s + 33 (cid:19) − (cid:18) s + 3 − e (cid:19)(cid:19) − (cid:18)(cid:18) s + 3 − e (cid:19) − (cid:18) s + 3 − e (cid:19)(cid:19) − (cid:18)(cid:18) s + 2 − e (cid:19) − (cid:18) s + 2 − e − b (cid:19)(cid:19) − b . Substituting s = − , we get τ d (Σ) e ( e + 1) / .Bounds in any codimension are calculated in the same way. Eachtime we will receive a polynomial in e .If the precise coefficients of this polynomial are not important, butonly its degree in e , it is claimed to be τ − . Indeed, let d be the gener-alized dimension of the system Σ , i.e. codimension τ of the polynomial ω Σ is equal to τ = m − − d . Apply to (5) operator ∆ d (while ∆ d ω Σ is a polynomial of degree zero, i.e. constant = τ d (Σ) ). Comparing thedegrees, we get that the degree w ′ = ∆ d w is less than τ . Let the mini-mizing coefficients of the polynomial w ′ equal to (0 , . . . , , b τ − , . . . , b ) .Replace in the resulting equation s variable on e and we have the fol-lowing relation: τ d (Σ) = (cid:18) s + τ + eτ (cid:19) − (cid:18) s + ττ (cid:19) − w ′ ( s ) . We use the definition (1) and get τ d (Σ) = (cid:18) s + τ + eτ (cid:19) − (cid:18) s + ττ (cid:19) − (7) X k = τ (cid:18) s + k − P kj = τ b j k (cid:19) − (cid:18) s + k − P k − j = τ b j k (cid:19)! . Denote by c i = P τ − j = i b j and rewrite equation (7) in this form: τ d (Σ) = (cid:18) s + τ + eτ (cid:19) − (cid:18) s + ττ (cid:19) − (8) X k = τ (cid:18)(cid:18) s + k − c k k (cid:19) − (cid:18) s + k − c k − k (cid:19)(cid:19) , Using identity (cid:18) s + k − − ak (cid:19) = (cid:18) s + k − ak (cid:19) − (cid:18) s + k − − ak − (cid:19) , transform the equation (8) to the form: τ d (Σ) = (cid:18) s + τ + eτ (cid:19) − (cid:18) s + ττ (cid:19) + X k = τ (cid:18) s + k − − c k − k (cid:19) +( s +1 − c ) . (9)Take ∆ τ − from both sides of the equality (9). Will have: s + 1 + e ) − s + 1) + ( s − c τ − ) + 1 , ЫрРәСЋРұРө c τ − = e . By inductionon i , we prove that for i < τ − it holds: c i = O ( e ( τ − − i ) . Let c j = O ( e ( τ − − j ) for all j : i j < τ − .Take ∆ i − from both sides of the equality (9) and get: (cid:18) s + τ − i + 1 + eτ − i + 1 (cid:19) − (cid:18) s + τ − i + 1 τ − i + 1 (cid:19) + i − X k = τ (cid:18) s + k − i − c k − k − i + 1 (cid:19) Substituting − instead of s , we get (cid:18) τ − i + eτ − i + 1 (cid:19) − (cid:18) τ − iτ − i + 1 (cid:19) + i +1 X k = τ (cid:18) k − i − − c k − k − i + 1 (cid:19) − c i − In the last sum we make the change j = k − i + 1 and get (cid:18) τ − i + eτ − i + 1 (cid:19) + j = τ +1 − i X j =2 (cid:18) j − c i + j − j (cid:19) − c i − . Now we have a formula expressing c i − through c i , ..., c τ − : c i − = τ − i +1 X j =2 O (cid:18)(cid:18) c i + j − j (cid:19)(cid:19) = τ − i +1 X j =2 O ( c ji + j − ) = τ − i +1 X j =2 O ( e j τ − i − j +1 ) = O ( e · τ − − i ) + τ − i +1 X j =3 O ( e j τ − i − j +1 ) = O ( e τ − i ) + τ − i +1 X j =3 O ( e j − · τ − i − j +1 ) = O ( e τ − i ) + τ − i X j =3 O ( e τ − i ) = O ( e τ − i ) (we used the inductive assumption and the fact that j j − for all j > ).Substituting s = − into the equation (9), we obtain τ d (Σ) = O ( c ) − c O (2 τ − ) . (cid:3) It is not known whether the resulting double exponential typical di-mension bound of graded ideal in a ring of generalized polynomials isbeing achieved. Note that it was proved in ([8]) that for degrees of ele-ments in the Gr¨obner basis of the polynomial ideal double exponentialbound is achieved from below. 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