Explicit Stillman bounds for all degrees
aa r X i v : . [ m a t h . A C ] S e p EXPLICIT STILLMAN BOUNDS FOR ALL DEGREES
GIULIO CAVIGLIA AND YIHUI LIANG
Abstract.
In 2016 Ananyan and Hochster proved Stillman’s conjecture by showing the ex-istence of a uniform upper bound for the projective dimension of all homogeneous ideals, inpolynomial rings over a field, generated by n forms of degree at most d . Explicit values of thebounds for forms of degrees 5 and higher are not yet known.The main result of this article is the construction of explicit such bounds, for all degrees d ,which behave like power towers of height d + d −
4. This is done by establishing a bound D ( k, d ), which controls the number of generators of a minimal prime over an ideal of a regularsequence of k or fewer forms of degree d , and supplementing it into Ananyan and Hochster’sproof in order to obtain a recurrence relation. Introduction
Let K be a field, R = K [ x , . . . , x N ] be a polynomial ring over K , and I be an ideal of R generated by n forms of degrees d , . . . , d n . We will denote the projective dimension of amodule over R by pd R ( M ). Stillman (see [21]) conjectured that the projective dimension of I can be bounded in terms of n and d , . . . , d n but independent of N . We will refer to suchbounds as Stillman bounds. Ananyan and Hochster were the first to give an affirmative answerto Stillman’s conjecture in [2], where they show the existence of Stillman bounds by proving theexistence of small subalgebras and small subalgebra bounds η B (defined in [2, Theorem B] orTheorem 2.2). Stillman’s conjecture was later reproved in [12] and [17], both using topologicalNoetherianity results from [13].With the existence proven, the next question is to find explicit Stillman bounds. Whilemany early and recent work [1] [3] [4] [5] [9] [16] [15] [18] [19] [20] has established Stillmanbounds for degree 4 or less, the question for degree 5 and higher remains untouched. We giveexplicit Stillman bounds for all degrees by proving explicit small subalgebra bounds η B througha recurrence relation in Theorem 3.7. In particular the Stillman bound we obtain is estimatedto be a power tower as follows: Theorem 4.3. If I is a homogeneous ideal in a polynomial ring R generated by n forms withmaximum degree d ≥ , then pd R ( R/I ) can be bounded by a power tower with base , height d + d − , and top exponent d + n + 3 . A key ingredient in our proof, borrowed from Ananyan-Hochster’s proof, is a bound called D ( k, d ), which controls the number of generators of a minimal prime over an ideal of a regularsequence of k or fewer polynomials with degree d . In Ananyan and Hochster’s inductive ar-gument, they only showed the existence of D ( k, d ). We make D ( k, d ) explicit in the followinglemma. Date : September 8, 2020.2010
Mathematics Subject Classification.
Primary 13D02, 13F20, 13P20.
Key words and phrases.
Stillman bound, projective dimension, regular sequence, R η . Lemma 3.3.
Let K be an algebraically closed field, P ⊂ K [ x , . . . , x N ] be a minimal primeof an ideal generated by a regular sequence of k or fewer forms of degree at most d . Then theminimal number of generators of P is bounded by D ( k, d ) = (2 d ) B ( k,d ) − . By inserting the above bound D ( k, d ) into the double-layered induction of [2], we obtain abound for η B and thus our Stillman bounds. Notice that in the above lemma, the assumptionof the field being algebraically closed is needed only for existence of B ( k, d ) as in [2, TheoremB], and the proof still works if we drop this assumption and replace B ( k, d ) by pd R ( P ) or N (see Remark 3.6).The paper is organized as follows. In Section 2, we first state the small subalgebra theoremsof [2] after giving their necessary definitions, then set up notations needed for the next section.In Section 3, we construct the bound D ( k, d ), discuss how we insert the bound D ( k, d ) intoAnanyan and Hochster’s proof, and conclude the section with a theorem establishing our boundsfor η B with a recurrence relation. In Section 3, we give an estimate of our bounds for η B .2. Notation
We first recall some definitions in [2, §
1] which are needed for stating the theorems in [2]. Let R = K [ x , . . . , x N ] be a polynomial ring over a field K . Let V be a finite dimensional gradedvector space of R, then we say V has dimension sequence δ = ( δ , . . . , δ d ) if V = V ⊕ · · · ⊕ V d as a direct sum of its graded components with dim K V i = δ i .A function of several variables is called ascending if it is increasing in any one variable whilethe other variables are fixed.A form F ∈ R has a k -collapse if it can be written as a graded combination of k or fewerforms of strictly smaller positive degree. We say F has strength k if it has a k + 1-collapse butno k -collapse. By convention we set the strength of a linear form to be + ∞ .A sequence of elements G , . . . , G s in a Noetherian ring R is a prime sequence (respectively,an R η -sequence ) if for 0 ≤ i ≤ s , R/ ( G , . . . , G i ) is a domain (respectively, satisfies the Serrecondition R η ). When R is a polynomial ring, for any η ≥ R η -sequence is a prime sequenceand hence a regular sequence.Our goal is to give explicit bounds to the functions η A and η B for any degree, which aredefined in Theorem A, Theorem B, and Corollary B of [2]. Theorem 2.1 (Ananyan-Hochster [2]) . There are ascending functions A = ( A , . . . , A d ) and,for every integer η ≥ , η A = ( η A , . . . , η A d ) from dimension sequences δ = ( δ , . . . , δ d ) ∈ N d to N d with the following property: For every algebraically closed field K and every positive integer N , if R = K [ x , . . . , x N ] is a polynomial ring, and V denotes a graded K -vector subspace of R of vector space dimension n with dimension sequence ( δ , . . . , δ d ) , such that for ≤ i ≤ d ,the strength of every nonzero element of V i is at least A i ( δ ) (respectively, η A i ( δ ) ), then everysequence of K -linearly independent forms in V is a regular sequence (respectively, is an R η -sequence). Theorem 2.2 (Ananyan-Hochster [2]) . There is an ascending function B from dimensionsequences δ = ( δ , . . . , δ d ) to Z + with the following property. If K is an algebraically closedfield and V is a finite-dimensional Z + -graded K -vector subspace of a polynomial ring R over K with dimension sequence δ , then V (and, hence, the K -subalgebra of R generated by V ) iscontained in a K -subalgebra of R generated by a regular sequence G , . . . , G s of forms of degree XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 3 at most d , where s ≤ B ( δ ) . Moreover, for every η ≥ there is such a function η B with theadditional property that every sequence consisting of linearly independent homogeneous linearcombinations of the elements in G , . . . , G s is an R η -sequence. The next corollary, as remarked in [2], follows immediately by taking η B ( n, d ) to be themaximum of η B ( δ ) over all dimension sequences with at most d entries and sum of entries atmost n . Corollary 2.3 (Ananyan-Hochster [2]) . There is an ascending function η B ( n, d ) , independent of K and N , such that for all polynomial rings R = K [ x , . . . , x N ] over an algebraically closed field K and all graded vector subspaces V of R of dimension at most n whose homogeneous elementshave positive degree at most d , the elements of V are contained in a subring K [ G , . . . , G B ] ,where B ≤ η B ( n, d ) and G , . . . , G B is an R η -sequence of forms of degree at most d . We next introduce notations that are needed for the proof of Lemma 3.2 and Lemma 3.3.For a finitely generated graded R -module M , let β ij ( M ) be the graded Betti numbers of M and β i ( M ) = P j β ij ( M ) be the i -th Betti number of M . The Castelnuovo-Mumford regularityof M is defined as reg( M ) =max i,j { j − i : β ij ( M ) = 0 } .Let J be a monomial ideal. We say J is strongly stable if for each monomial u of J , x i | u implies x j u/x i ∈ I for each j < i . Let G ( J ) be the set of minimal monomial generators of J and D ( J ) be the largest degree of monomials in G ( J ). If u is a monomial, let m ( u ) := max { i : x i | u } .By the Eliahou-Kervaire resolution in [14], if J is strongly stable then β i ( J ) = P u ∈ G ( J ) (cid:0) m ( u ) − i (cid:1) .Let I be a monomial ideal in K [ x , . . . , x N ] where K is an infinite field, we may assume I is generated by monic monomials, if K ′ is any other field then let I K ′ be the ideal generatedby the image of these monomials in K ′ [ x , . . . , x N ]. Let gin rlex ( I ) be the generic initial ideal of I with respect to the degree reverse lexicographical order. The zero-generic initial ideal of I with respect to the degree reverse lexicographical order is defined to beGin ( I ) := (gin rlex ((gin rlex ( I )) Q )) K . The zero-generic initial ideal is explored in more details in [8]. We need this notion in § Procedures to realize the bound via a recursive algorithm
We start this section by constructing a bound, denoted D ( k, d ) in [2], for the number ofgenerators of a minimal prime of an ideal generated by a regular sequence of k or fewer forms ofdegree d . We note that this bound is independent of the number of variables in the polynomialring, which is necessary for our purpose. The following lemma, see [10, Theorem 27], tells usthat the above minimal prime can be written as the ideal of the regular sequence colon by aform with bounded degree, which is key to the proof of Lemma 3.2. Lemma 3.1 (Chardin [10]) . Let P ⊂ K [ x , . . . , x N ] be a minimal prime of an ideal generatedby a homogeneous regular sequence f , . . . , f k of degrees d , . . . , d k . There exists a form f ofdegree at most d + · · · + d k − k such that P = ( f , . . . , f k ) : ( f ) . The next two lemmas justify the fact that we can choose D ( k, d ) to be (2 d ) B ( k,d ) − . Assumingthe number of variables is known, we show in Lemma 3.2 how to bound the minimal number XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 4 of generators of a particular kind of colon ideals as in Lemma 3.1. In pursuance of an optimalbound, we extensively apply results and proofs of [7] and [8]. For an alternate way to obtaina value for D ( k, d ), we refer the reader to Remark 3.5, which could still be used to constructexplicit but larger bounds for η B ( δ ). Lemma 3.2.
Let I ⊂ K [ x , . . . , x B +1 ] be an ideal generated by a regular sequence of c formsof degree at most d , and f be a form of degree at most cd − c . Then the minimal number ofgenerators of I : ( f ) is bounded by β ( I : ( f )) ≤ B ( d + 1)(2 d ) B Y i =3 (( d + 2 d − i − + 1) + 1 ≤ (2 d ) B − , where the last inequality holds if we further assume B ≥ , or B ≥ and d ≥ .Proof. If c = 1 then β ( I : ( f )) = 1, so assume c ≥
2. After a faithfully flat base change,we may assume K is infinite. Denote R := K [ x , . . . , x B +1 ] and let f , . . . , f c be the regularsequence with deg( f i ) ≤ d . Consider the exact sequence0 → R/ ( I : ( f )) f −−→ R/I −→ R/ ( I + ( f )) → . Since c ≤ β ( I + ( f )) ≤ c + 1, using the long exact sequence of Tor Ri ( − , K ) induced from theabove short exact sequence, we get β ( I : ( f )) ≤ β ( I + ( f )) + 1.With the notations of Section 2, let J := Gin ( I + ( f )). Denote R [ i ] := K [ x , . . . , x i ].Let ( I + ( f )) h i i denote its image in R/ ( l B +1 , . . . , l i +1 ) ∼ = R [ i ] where l B +1 , . . . , l i +1 are generallinear forms, let J [ i ] denote J ∩ R [ i ] . By [1, Proposition 2.2], J is strongly stable with β ( I +( f )) ≤ β ( J ). So by [14] and [7, Proposition 1.6], β ( J ) = P u ∈ G ( J ) ( m ( u ) − ≤ B · | G ( J ) | ≤ B Q Bi =1 ( D ( J [ i ] ) + 1). Using [8, Theorem 2.20], we can get D ( J [ i ] ) ≤ reg(( I + ( f )) h i i ) for all i .Notice that reg(( I + ( f )) h i i ) ≤ id − i + 1 for all i ≤ c , because m id − i +1 h i i ⊆ ( I + ( f )) h i i where m = ( x , . . . , x B +1 ).To bound reg(( I + ( f )) h i i ) for i ≥ c + 1, we follow the proof of [7, Theorem 2.4 and Corollary2.6]. Let λ ( M ) denote the length of an Artinian module M . Using the same proof of [7,Theorem 2.4], we can getreg(( I + ( f )) h i i ) ≤ max { d, cd − c, reg(( I + ( f )) h i − i ) } + λ (cid:18) R [ c ] ( I + ( f )) h c i (cid:19) i Y j = c +2 reg(( I + ( f )) h j − i ) ≤ max { d, cd − c, reg(( I + ( f )) h i − i ) } + d c i − Y j = c +1 reg(( I + ( f )) h j i ) . (3.1)The last inequality holds since R [ c ] / ( I + ( f )) h c i is a quotient ring of R [ c ] / ( g , . . . , g c ), where g , . . . , g c is the image of f , . . . , f c in R [ c ] and is a regular sequence with deg( g i ) ≤ d .Now we use (3.1) recursively to bound reg(( I + ( f )) h i i ) for i ≥ c + 1. Set B := cd − c + 1,recall that B bounds reg(( I + ( f )) h c i ). Apply 3.1 to ( I + ( f )) h c +1 i to get reg(( I + ( f )) h c +1 i ) ≤ cd − c + 1 + d c =: B . For j ≥
2, we set B j := B j − + d c Q j − k =1 B k ≤ ( B j − ) ≤ ( B ) j − . Hencefor all i ≥ c + 1, reg(( I + ( f )) h i i ) ≤ B i − c ≤ ( cd − c + 1 + d c ) i − c − ≤ ( d + 2 d − i − , where thelast inequality holds since the second last bound is decreasing as a function of c and c ≥ XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 5 If B ≥
4, then B ≤ B − . Combining all the previous inequalities, we get β ( I : ( f )) ≤ B ( d + 1)(2 d ) B Y i =3 (( d + 2 d − i − + 1) + 1 ≤ B − ( d + 1)(2 d ) B Y i =3 ( d + 2 d ) i − ≤ (2 d )(2 d ) B Y i =3 (2 d + 4 d ) i − ≤ (2 d ) B Y i =3 (2 d ) i − = (2 d ) B − . If B = 3 and d ≥
3, one easily checks that β ( I : ( f )) ≤ d + 1)(2 d )( d + 2 d ) + 1 ≤ (2 d ) . (cid:3) Lemma 3.3 constructs our value (2 d ) B ( k,d ) − for the bound D ( k, d ) by passing to a polynomialsubring with at most B ( k, d ) + 1 many variables first, using Corollary 2.3, then combining theresult of Lemma 3.1 and Lemma 3.2. Lemma 3.3.
Let K be an algebraically closed field, P ⊂ K [ x , . . . , x N ] be a minimal primeof an ideal generated by a regular sequence of k or fewer forms of degree at most d . Then theminimal number of generators of P is bounded by β ( P ) ≤ (2 d ) B ( k,d ) − . Proof.
Let f , . . . , f c be the regular sequence with c ≤ k and deg( f i ) ≤ d , let I be the ideal itgenerates. By Corollary 2.3, there exists a prime sequence G , . . . , G s with s ≤ B ( k, d ) suchthat f , . . . , f c ∈ K [ G , . . . , G s ]. Denote R = K [ x , . . . , x N ] and S = K [ G , . . . , G s ]. Thenpd R ( R/I ) = c ≤ s and pd S ( S/P ∩ S ) ≤ s . Notice that R is a free and thus faithfully flatmodule over S since we can extend G , . . . , G s to a maximal regular sequence G , . . . , G N in R to get free extensions K [ G , . . . , G s ] ֒ −→ K [ G , . . . , G N ] and K [ G , . . . , G N ] ֒ −→ K [ x , . . . , x N ].Consequently we get pd R ( R/P ) ≤ s once we have shown P = ( P ∩ S ) R . By faithfully flatness f , . . . , f c ∈ P ∩ S remains a regular sequence in S and so c = ht P ≥ ht ( P ∩ S ) R = ht P ∩ S ≥ c .Now by [2, Corollary 2.9], ( P ∩ S ) R is a prime ideal, therefore P = ( P ∩ S ) R .By 3.1, there exists a form f ∈ R of degree at most cd − c such that P = I : ( f ). Considerthe exact sequence 0 → R/P f −−→ R/I −→ R/ ( I + ( f )) → . (3.2)It follows that pd R ( R/ ( I + ( f ))) ≤ s + 1. Then depth R/ ( I + ( f )) ≥ N − ( s + 1) by theAuslander-Buchsbaum formula. Let l s +2 , . . . , l N ∈ R be a sequence of linear forms regularon R/P , R/ ( I + ( f )), and R/ ( f , . . . , f c ). Fix a graded isomorphism from R/ ( l s +2 , . . . , l N )to K [ x , . . . , x s +1 ], let R denote K [ x , . . . , x s +1 ] and “ ” denote the image of polynomials orideals of R in R . Notice that β ( P ) = β ( P ). Since l s +2 , . . . , l N is a regular sequence on R/ ( I + ( f )), the short exact sequence in (3.2) remains exact after tensoring with R . It followsthat P = I : ( f ). Notice that f , . . . , f c is a regular sequence in R , so we can apply Lemma 3.2to get β ( P ) ≤ (2 d ) B ( k,d ) − for B ( k, d ) ≥
4, or B ( k, d ) = 3 and d ≥ B ( k, d ) = 2, it is clear that β ( P ) = β ( P ∩ S ) ≤
2. If B ( k, d ) = 3 and d = 2, we recallthat pd R ( R/P ) ≤ s ≤
3. After moding out N − P isany ideal containing a regular sequence of 2 forms of degree at most 2 in 3 variables. ApplyingRemark 3.4 below, we get β ( P ) ≤ (cid:3) XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 6
Remark
Number of generators for curves and points in P ) . Let I be any ideal in K [ x , x , x ]containing a regular sequence of 2 forms of degree d and d , we claim that I has at most d d +2minimal generators. Assume d ≤ d , since the Eisenbud-Greene-Harris conjecture hold for reg-ular sequence of length 2 (see for instantce [6]), there exists a lex-plus-power ideal J , writtenas ( x d , x d ) plus a lexsegment ideal, whose Hilbert function is the same as the one of I. Fur-thermore J has at least as many minimal generators as I . Finally notice that a minimal setof monomial generators for J must be a subset of { x d , x d } ∪ S ≤ a ≤ d − , ≤ b ≤ d − { x a x b x c a,b } forsome exponents c a,b ≥ Remark . By using known results, there is a quick and simple way to obtain a worse upperbound than the one derived from Lemma 3.3. The outline is the following. First one uses thebound on the degrees of the generators of initial ideals given in [11, Corollary 3.6] and the proofof [11, Corollary 3.7] to bound the degrees of the generators of P = I : ( f ). When d ≥
2, thebound one gets is ( d c ( cd − c )) s +1 − c ≤ ( d (2 d − B ( k,d ) − =: D . Hence, by linear independency,the number of minimal homogeneous generators of P is at most (cid:0) B ( k,d )+ DD (cid:1) . Remark . In the proof of Lemma 3.3, notice that we could replace B ( k, d ) by the projec-tive dimension of the minimal prime P . The same proof will give us the bound: β ( P ) ≤ (2 d ) pd( P ) − ≤ (2 d ) N − . We conclude this section by presenting a recursive formula that computes η B ( δ ), where δ = ( δ , . . . , δ d ) is a dimension sequence (see Theorem 3.7). The theoretical proof of [2, § d , can be easily made into an algorithmonce we insert the bound obtained in Lemma 3.3.Denote η A ( i ) = B ( D ( η + 1 , i − , i −
1) + 1. Then by [2, Proposition 2.6 and Theorem A],we have η A i ( δ ) = η A ( i ) + 3( P dj =1 δ j − η B from η A , which we will describe briefly as follows.Let V be any vector space with dimension sequence δ , if for all such V , the strength of everynonzero element of V i is at least η A ( δ ), then let η B ( δ ) = P di =1 δ i . Otherwise there exists a V and a degree i for which an element of V i has an η A i ( δ )-collapse. In this case set η B ( δ ) =max δ ′ { η B ( δ ′ ) } , where δ ′ run through all dimension sequences derived from δ by keeping δ j unchanged for j > i , decreasing δ i by 1, and increasing the δ j ’s for j < i by a total of 2 · η A d ( δ ).Notice that if d = 1, η B (( δ )) = δ trivially satisfies 2.2.By considering the worst case scenario in the algorithm described above, we may define η B recursively as η B ( δ , . . . , δ d ) = η B ( δ , . . . , δ d − , δ d − + 2 η A d ( δ ) , δ d − η B ( δ , . . . , δ d − , a i , δ d − i ) , where a i satisfies the recurrence relation a = δ d − , a i = a i − + 2( η A ( d ) + 3( a i − + δ + · · · + δ d − + δ d − ( i − − a i − + 2 η A ( d ) + 6( δ + · · · + δ d − + δ d ) − i. Notice that a i = 7 i ( δ + · · · + δ d + η A ( d ) − ) + i − ( δ + · · · + δ d − + δ d ) − η A ( d ) + isa solution of the recurrence relation. Our goal is to bring down all the degree d elements to XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 7 degree d −
1, and so we compute a δ d to get η B ( δ , . . . , δ d ) = η B ( δ , . . . , δ d − , a δ d ,
0) = η B ( δ , . . . , δ d − , δ d ( δ + · · · + δ d + 13 η A ( d ) −
76 ) − ( δ + · · · + δ d − ) − η A ( d ) + 76 ) . With our recursive definition we have η B ( n, d ) = η B ( δ ) where δ = (0 , · · · , , n ) has d entries.The algorithm finishes by iterating the above process until all forms are in degree 1. Wesummarize the above discussion into the following theorem. Theorem 3.7.
Under the setting of Theorem 2.2, we have the bound η B ( δ ) = b d − where b d − satisfies the following recurrence relation: b i = 7 b i − (cid:18) B ( D ( η + 1 , d − i ) , d − i ) + b i − + δ + · · · + δ d − i − (cid:19) − B ( D ( η + 1 , d − i ) , d − i ) − ( δ + · · · + δ d − i − ) + 56 . with b = δ d , D ( η + 1 , d − i ) = [2( d − i )] B ( η +1 ,d − i ) − if d − i > , D ( η + 1 ,
1) = η + 1 , and η B (( δ )) = δ . Estimate of the small subalgebra bound
In this section, we give an estimate for the bound η B ( n, d ) in Corollary 2.3. Notice that thisalso estimates the bound η B ( δ ) in Theorem 2.2 since if δ is a dimension sequence with d entrieswhose sum equals to n , then η B ( δ ) ≤ η B ( n, d ).Unfortunately a formula that represents the actual value of the recursion described in Theo-rem 3.7 will be a very complicated, hard to read formula that behaves like a power tower. Wedecided, for the sake of clarity, to push all the exponents up in order to obtain a reasonableupper bound written as a power tower with linear top exponent depending on d, n, η . We focusnot on the optimal value of the top exponent, but on the height of power tower.We denote a power tower with base a , height n , and top exponent x by Notation 4.1.
Let exp na ( x ) = a a ·· ax with n a ’s. The result of our estimate, whose details are at the end of the section, is as follows.
Proposition 4.2.
Assume d ≥ , then η B ( n, d ) defined in Corollary 2.3 can be bounded by η B ( n, d ) ≤ exp d + d − ( d + n + η + 2) . In particular when η = 1, B ( n, d ) gives a bound for the projective dimension. Theorem 4.3. If I is a homogeneous ideal in a polynomial ring R generated by n forms withmaximum degree d ≥ , then pd ( R/I ) can be bounded by a power tower with base , height d + d − , and top exponent d + n + 3 . For the rest of this section, we compute the estimate in 4.2. During the computations, wewill push up the exponents repeatedly, by which we mean:If a ≥ , x, y, z ≥ , and x + y ≤ a y ′ , then a ya z + x ≤ a ya z + x ≤ a ( x + y ) a z ≤ a a y ′ a z = a a y ′ + z . XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 8
For the computations below, we work with the recurrence in Theorem 3.7. Recall that η B ( n, d ) = η B ( δ ) where δ = (0 , · · · , , n ) has d entries, and η B ( n, d ) = b d − with the b i ’s definedin Theorem 3.7. We assume d ≥ b i ’s in Theorem 3.7. b i = 7 b i − (cid:18) B ( D ( η + 1 , d − i ) , d − i ) + b i − − (cid:19) − B ( D ( η + 1 , d − i ) , d − i ) + 56 ≤ b i − ( 13 B ( D ( η + 1 , d − i ) , d − i ) + b i − ) ≤ b i − ( B ( D ( η + 1 , d − i ) , d − i )) . (4.1)We push up the exponents appearing in η B ( n, d ) = b d − .Notice that 2 B ( D ( η + 1 , d − i ) , d − i ) ≤ B ( D ( η +1 ,d − i ) ,d − i ) and B ( D ( η + 1 , d − , d − ≥ B ( D ( η + 1 , d − i ) , d − i ) for all i ≥ η B ( n, d ) = b d − ≤ B ( D ( η + 1 , , B ( D ( η +1 , , ··· B ( D ( η +1 ,d − ,d − n ≤ ( η + 1)exp d − ( B ( D ( η + 1 , ,
2) + · · · + B ( D ( η + 1 , d − , d −
1) + 2 n ) ≤ ( η + 1)exp d − (( d − B ( D ( η + 1 , d − , d −
1) + 2 n ) . (4.2)Apply 4.2 to B ( D ( η + 1 , d − , d − B ( D ( η + 1 , d − , d − ≤ d − (( d − B ( D (4 , d − , d −
2) + 2 D ( η + 1 , d − B ( D (4 , d − , d − , B ( D (4 , d − , d − , . . . , B ( D (4 , ,
2) and push up allthe exponents, we get: B ( D (4 , d − , d − ≤ d − (( d − B ( D (4 , d − , d −
3) + 2 D (4 , d − ≤ exp d − (( d − B ( D (4 , d − , d −
3) + 2 D (4 , d −
2) + 1) ≤ exp P d − j =1 j ( d − X j =2 j + 2( d − D (4 , d −
2) + d − d − d +62 (2( d − D (4 , d −
2) + 12 d − d + 2) . (4.4)Now we combine the above inequalities. The first inequality below follows from 4.2, the secondfrom 4.3, the third from 4( d − ≤ d − , the fourth from 4.4 and ( d −
3) + ( d − ≤ d − . η B ( n, d ) ≤ ( η + 1)exp d − (( d − B ( D ( η + 1 , d − , d −
1) + 2 n ) ≤ ( η + 1)exp d − (( d − d − (( d − B ( D (4 , d − , d −
2) + 2 D ( η + 1 , d −
1) + n )) ≤ ( η + 1)exp d − d − (( d − B ( D (4 , d − , d −
2) + 2 D ( η + 1 , d −
1) + n + ( d − ≤ ( η + 1)exp d − d − d +62 (2( d − D (4 , d −
2) + 12 d − d + 2 + D ( η + 1 , d −
1) + n + 12 d − η + 1)exp d − d (2( d − D (4 , d −
2) + D ( η + 1 , d −
1) + n + 12 d − d + 1) . (4.5) XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 9
In particular when η = 1: B ( n, d ) ≤ exp d − d (2( d − D (4 , d −
2) + D (2 , d −
1) + n + 12 d − d + 2) ≤ exp d − d ((2 d − D (4 , d −
1) + n + 12 d − d + 2) . We bound D ( η + 1 , d −
1) as: D ( η + 1 , d −
1) = [2( d − B ( η +1 ,d − − ≤ [7 d − ] exp d − d +227 ((2 d − D (4 ,d − η + 12 d − d + 112 ) ≤ exp d − d +62 ((2 d − D (4 , d −
2) + η + 12 d − d + 92 ) . (4.6)To estimate D (4 , d − ≤ j ≤ d −
2, we have: D (4 , j ) = [8 j ] B (4 ,j ) − ≤ [7 j ] exp j − j
27 ((2 j − D (4 ,j − j − j +2) ≤ exp j − j +42 ((2 j − D (4 , j −
1) + 12 j − j + 6) . (4.7)And when j = 2, we get the bound below. Notice that we can use the first line of 4.1 tobound B (4 ,
2) = 7 ( · − ) − · ≤ · . D (4 ,
2) = [4] B (4 , − ≤ [4] · − ≤ exp (5) . Apply 4.7 recursively, use the inequality P d − j =3 (2 j −
5) + ( j − j + 6) ≤ exp ( d − D (4 , d − ≤ exp P d − j =2 j − j +42 ( 12 d − d − d d − ( 12 d + 3) . (4.8)We combine the previous inequalities to get our final bound below. Notice that the firstinequality follows from 4.5, the second from 4.6, the fourth from 4.8 and the inequality d − d + + (2 d − ≤ exp ( d − η B ( n, d ) ≤ exp d − d (2( d − D (4 , d −
2) + D ( η + 1 , d −
1) + n + 12 d − d + 2 + η ) ≤ exp d − d +37 ( d − d − D (4 , d −
2) + η + 12 d − d + 92+ n + 12 d − d + 2 + η )= exp d − d +37 ((2 d − D (4 , d −
2) + 2 η + n + d − d + 72 ) ≤ exp d − d +3+ d − d d − ( 12 d + 3 + η + n + 12 d − d + d − ( d + 2 + n + η ) . In particular when η = 1, we get B ( n, d ) ≤ exp d + d − ( d + n + 3). XPLICIT STILLMAN BOUNDS FOR ALL DEGREES 10
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