Semi-invariants of binary forms and symmetrized graph-monomials
SSemi-invariants of binary forms and symmetrizedgraph-monomials
Shashikant Mulay
Department of Mathematics, University of TennesseeKnoxville, TN 37996 U. S. A.; e-mail: [email protected] 17, 2019
Abstract : This article provides a method for constructing invariants and semi-invariants of a binary N -ic form over a field k characteristics 0 or p > N . Apractical and broadly applicable sufficient condition for ensuring nontriviality ofthe symmetrization of a graph-monomial is established. This allows construc-tion of infinite families of invariants (especially, skew-invariants) and families of k -linearly independent semi-invariants. These constructions are very useful inthe quantum physics of Fermions. Additionally, they permit us to establish anew polynomial-type lower bound on the coefficient of q w in ( q − (cid:0) N + dd (cid:1) q forall sufficiently large integers d and w ≤ N d/ Keywords : Symmetrized graph-monomials, Semi-invariants of binary forms.
MSC Classifications : 05E05, 13A50.Fix an integer N ≥
2. Let k be a field of characteristic either 0 or strictlygreater than N . Let X , Y , t , z , . . . , z N be indeterminates. Let E ( t ) , . . . , E N ( t )and f ( X + t ) be the polynomials defined by f ( X + t ) := N (cid:89) i =1 ( X + z i + t ) =: X N + N (cid:88) i =1 E i ( t ) X N − i . For 1 ≤ i ≤ N , let e i := E i (0). Then, f ( X ) = X N + e X N − + · · · + e N . A poly-nomial P ( e , . . . , e N ) ∈ k [ e , . . . , e N ] is said to be translation invariant provided P ( E ( t ) , . . . , E N ( t )) = P ( e , . . . , e N ). It is a (well known) simple exercise to ver-ify that the subring k [ y , . . . , y N − ] of k [ e , . . . , e N ], where y i := E i ( − e /N ) for1 ≤ i ≤ N , is the ring of all translation invariant members of k [ e , . . . , e N ].Furthermore, we have k [ y , . . . , y N − ] = k [ e , . . . , e N ] ∩ k [ z − z , . . . , z − z N ]( e.g. , see Ch. 2, Theorem 1 of [11]). A polynomial h ∈ k [ e , . . . , e N ] is saidto be homogeneous of weight w provided as a polynomial in z , . . . , z N , h ishomogeneous of degree w . Note that y i is homogeneous of weight i + 1 for1 ≤ i ≤ N . Next, consider the (generic) binary form F := (cid:80) a i X i Y N − i ofdegree N where a is an indeterminate and a i := a e i for 1 ≤ i ≤ N . A semi-invariant of F of degree d and weight w is a polynomial Q ∈ k [ a , a , . . . , a N ]such that Q = a d P ( e , . . . , e N ) where P ( e , . . . , e N ) is translation invariant, ho-mogeneous of weight w and has total degree ≤ d in e , . . . , e N . For 0 ≤ i ≤ N ,the weight of a i is defined to be i . Then, note that Q is homogeneous of degree d and weight w in a , . . . , a N . An invariant of F of degree d is a semi-invariantof F of degree d and weight N d/
2. For a fixed N , the set of semi-invariants a r X i v : . [ m a t h . A C ] M a y of the binary N -ic F ) of degree d and weight w form a finite dimensional k -linear subspace of k [ a , a , . . . , a N ]. This subspace is known to be trivial unless2 w ≤ N d . Provided char k = 0 and 2 w ≤ N d , a theorem of Cayley-Sylvesterproves that the dimension of the aforementioned space of semi-invariants of de-gree d and weight w is the coefficient of q w in ( q − (cid:0) N + dd (cid:1) q where (cid:0) N + dd (cid:1) q is the q -binomial coefficient (see [6], [18] or Theorem 5 of [11]). Let p w ( N, d )denote the coefficient of q w in q w in (cid:0) N + dd (cid:1) q . Then, p w ( N, d ) is the numberof integer-partitions of w in at most N parts with each part ≤ d . As a corol-lary of the Cayeley-Sylvester theorem, we then have p w ( N, d ) ≥ p w − ( N, d )for 2 ≤ w ≤ N d/
2; this establishes unimodality of the coefficients of (cid:0) N + dd (cid:1) q .Since p w ( N, d ) − p w − ( N, d ) are the dimensions of spaces of semi-invariants, itis natural to investigate explicit (lower, upper) bounds on them. Recently, someinteresting lower bounds on p w ( N, d ) − p w − ( N, d ) have come to light (see [4],[12], [19] and their references). This article has two objectives: provide explicitmethods of constructing a class of k -linearly independent semi-invariants andobtain a new lower bound on p w ( N, d ) − p w − ( N, d ) for certain pairs ( w, d ). Thenon-trivial lower bounds of [4], [12] and [19] are valid for min { N, d } ≥ d and w , they do not depend on ( w, d ). In contrast,our lower bounds (see Theorem 3) are polynomials in w for all ( N, d ); Example3, 4 and Remark 5 appearing at the end of the article present a more detailedcomparison. In this article, we investigate the algebra of semi-invariants; notthe combinatorics of q -binomial coefficients. In the rest of the introduction, wedescribe our motivation for, and our method of, constructing semi-invariants ofa binary N -ic form.Ever since the theory of invariants of binary forms was founded, invariant-theorists have explored and devised methods for writing down concrete invari-ants; however, each of these methods has its own shortcomings. The ‘symbolicmethod’ of classical invariant theory (see [3], [6], [7], [9]) provides an easy recipefor formulating symbolic expressions that yield invariants and semi-invariants.But, without full expansion (or un-symbolization) one does not know whethera given symbolic expression yields a nonzero semi-invariant. Here we prefer theother method, i.e. , the method of symmetrized graph-monomials. This too wasknown to classical invariant theorists (see [13], [14], [17]). It poses the problemof finding a useful criterion to determine nonzero-ness of the symmetrization.Historically, Sylvester and Petersen considered this problem; in fact, Petersenformulated a sufficient (but not necessary) condition that ensures zero-ness ofthe symmetrization. For a detailed historical sketch of this topic, we refer thereader to [16]. In [16], nonzero-ness of the symmetrization of a graph-monomialis shown to be equivalent to certain properties of the orientations and the orien-tation preserving graph-automorphisms of the underlying graph; but as mattersstand, verification of these properties is as forbidding as is a brute force compu-tation of the desired symmetrization. Our interest in construction , as opposedto existence , of invariants and semi-invariants stems primarily from the needto obtain explicitly described trial wave functions for systems of N stronglycorrelated Fermions in fractional quantum Hall state. Such a trial wave func-2ion is essentially determined by a so called correlation function . The intuitiveapproach of physics presents such a correlation function as a symmetrizationof a monomial obtained from the graph of correlations representing allowedstrong interactions between N Fermions. It so happens that this correlationfunction turns out to be a semi-invariant (an invariant in certain cases), of abinary N -ic form. In this article, we establish an easy to use yet broadly ap-plicable sufficient criterion (see Theorem 1) for non-triviality of a symmetrizedgraph-monomial. Besides enabling explicit constructions of the desired trialwave functions, Theorem 1 is also interesting from a purely invariant theoreticpoint of view. Following Theorem 1, we exhibit a sample of its applications (seeTheorem 2, Theorem 3)..A multigraph is a graph in which multiple edges are allowed between thesame two vertices of the graph. Consider a loopless undirected multigraph Γon finitely many (at least two) vertices labeled 1 , , . . . , N ; multigraph Γ is saidto be d -regular provided each vertex of Γ has the same degree d . In the figuresbelow, Γ is seen to be a 2-regular multigraph and the multigraphs Γ , Γ bothare 3-regular. l lllll Figure 1: Γ l lll Figure 2: Γ l lll Figure 3: Γ Let ε (Γ , i, j ) be the number of edges in Γ connecting vertex i to vertex j . The graph-monomial of Γ, denoted by µ (Γ), is the polynomial in z , . . . , z N definedby µ (Γ) := (cid:89) ≤ i N, L, D ). Likewise, given Γ , Γ (cid:48) ∈ C ( N, L, d ),it is of interest to know when g (Γ) is (or is not) a constant multiple of g (Γ (cid:48) ).Without digressing into deeper physics, we simply refer the reader to [2], [10],[11] and [15]. Using a weak corollary of Theorem 1 of this article (also, Theo-rem 1 of [9]), we have explicitly constructed trial wave functions for the minimalIQL configurations of N Fermions in a Jain state with filling factor < / g (Γ). There is nothing akin toTheorem 1 in the existing literature. Whenever Theorem 1 is applicable to evena single member of C ( N, L, d ), it readily yields an upper bound on p ( N, L, d ).Our proof of Theorem 1 is purely algebraic in nature; so, the edge-function (orthe edge-matrix) of a multigraph is of key importance in the proof. In Theo-rem 1 we consider only those multigraphs Γ that can be partitioned into twoor more sub-multigraphs Γ , . . . , Γ m such that each g (Γ i ) is nonzero (in par-ticular, if Γ i has no edges) and the inter-edges between pairs Γ i , Γ j are more‘dominating’ (in a specific way) than the intra-edges within each Γ i . Using The-orem 1, we are able to construct several infinite families of invariants (includingskew-invariants, see Theorem 2) as well as families of k -linearly independentsemi-invariants of a binary N -ic form over k (see Theorem 3). At its core,our approach has its source in [1]; this is very philosophical and hence almostimpossible to articulate. In closing, we share our optimism that there is a gen-eralization of Theorem 1 yet to be discovered, that will allow construction of all4emi-invariants as symmetrized-graph-monomials.In what follows, N is tacitly assumed to be an integer ≥ k denotes a fieldand z , . . . , z N are indeterminates. We let z stand either for ( z , . . . , z N ) or theset { z , . . . , z N } . It is tacitly assumed that either k has characteristic 0 or thecharacteristic of k is > N . As usual, given a positive integer n , S n denotes thegroup of all permutations of the set { , . . . , n } . Definitions : Let m and n be positive integers.1. Let Symm N : k [ z ] → k [ z ] be the Symmetrization operator defined by Symm N ( f ) := (cid:88) σ ∈ S N f ( z σ (1) , . . . , z σ ( N ) ) .f ∈ k [ z ] is said to be symmetric provided f ( z σ (1) , . . . , z σ ( N ) ) = f ( z , . . . , z N ) for all σ ∈ S N .2. For an m × n matrix A := [ a ij ], let r i ( A ) := a i + · · · + a in (the sum ofthe entries in the i -th row of A ) for 1 ≤ i ≤ m and let (cid:107) A (cid:107) := r ( A ) + · · · + r m ( A ) = m (cid:88) i =1 n (cid:88) j =1 a ij . 3. Let E ( N ) denote the set of all N × N symmetric matrices A := [ a ij ] suchthat each a ij is a nonnegative integer and a ii = 0 for 1 ≤ i ≤ N .4. Given an integer d , by E ( N, d ) we denote the subset of A ∈ E ( N ) suchthat r i ( A ) = d for 1 ≤ i ≤ N , i.e. , each row-sum of A is exactly d .5. For an N × N matrix A := [ a ij ], let δ ( z, A ) := (cid:89) ≤ i 6. Let D ( m,n ) := [( c ij ] be the m × n matrix such that c ii := (cid:40) i = j ,1 if i (cid:54) = j .By D n , we mean D ( n,n ) . In particular, D = 0. Lemma 1 : Let n be a positive integer. For 1 ≤ i ≤ n , let g i ∈ Q ( z ). Then g + g + · · · + g n = 0 if and only if g i = 0 for 1 ≤ i ≤ n . In particular, given a0 (cid:54) = g ∈ Q ( z , . . . , z N ) and a nonempty subset S ⊆ S N , we have (cid:88) σ ∈ S g ( z σ (1) , . . . , z σ ( N ) ) (cid:54) = 0 . roof : With the notation of (i), assume that g (cid:54) = 0. Let h := g + g + · · · + g n .For 1 ≤ i ≤ n , let p i , q i ∈ Q [ z , . . . , z N ] be polynomials such that g i q i = p i and q i (cid:54) = 0. Note that, g (cid:54) = 0 implies p (cid:54) = 0. Now since f := p q q · · · q n isa nonzero polynomial with coefficients in Q , there exists ( a , . . . , a N ) ∈ Q N such that f ( a , . . . , a N ) (cid:54) = 0. Fix such an N -tuple ( a , . . . , a N ) and let c i := g i ( a , . . . , a N ) for 1 ≤ i ≤ n . Then, c (cid:54) = 0 and c i ∈ Q for 1 ≤ i ≤ n . Since c > c + · · · + c n ) ≥ 0, we have h ( a , . . . , a N ) > 0. This proves the firstclaim of (i); the second claim of (i) easily follows. Assertion (ii) readily followsfrom (i). (cid:3) Definitions :1. For B ⊆ { , , . . . , N } , let π ( B ) := { ( i, j ) ∈ B × B | i < j } . By abuse of notation, π ( B ) is also identified as the set of all 2-elementsubsets of B . The set π ( { , . . . , N } ) is denoted by π [ N ].2. Given C ⊆ π [ N ] and a function ε : C → N , the image of ( i, j ) ∈ C via ε is denoted by ε ( i, j ). An integer w ∈ N is identified with the constantfunction C → N such that ( i, j ) → w for all ( i, j ) ∈ C .3. Given C ⊆ π [ N ] and a function ε : C → N , define v ( z, C, ε ) := (cid:89) ( i,j ) ∈ C ( z i − z j ) ε ( i,j ) with the understanding that v ( z, ∅ , ε ) = 1. Remark 1 : There is an obvious bijective correspondence ε ↔ [ a ij ] between theset of functions ε : π [ N ] → N and the set E ( N ), given by a ij = ε ( i, j ) for 1 ≤ i < j ≤ N .Suppose m ≤ m ≤ · · · ≤ m q is a partition of N and M ∈ E ( N ). Consider M as a q × q block-matrix [ M rs ], where M rs has size m r × m s for 1 ≤ r, s ≤ q .View M as the sum M ∗ + M ∗∗ , where M ∗ is the q × q block-diagonal matrixhaving M rr as its r -th diagonal block and where M ∗∗ is the q × q block-matrixwhose diagonal blocks are zero-matrices. Clearly, M ∗ and M ∗∗ both are in E ( N ) and M rr ∈ E ( m r ) for 1 ≤ r ≤ q . Definitions : Let the notation be as above.1. For 1 ≤ r ≤ q , define A r := { i + m + · · · + m r − | ≤ i ≤ m r } . 6. For 1 ≤ r ≤ q , let G r denote the group of permutations of the set A r .3. Define π := (cid:91) ≤ r 4. For 1 ≤ r ≤ q and ( i, j ) ∈ π ( A r ), let ε r ( i, j ) denote the ij -th entry of M ∗ .5. For 1 ≤ r ≤ q , define δ r ( M ∗ ) := Symm m r ( v ( z, π ( A r ) , ε r )) . 6. For ( i, j ) ∈ π [ N ], let ε ( i, j ) denote the ij -th entry of M ∗∗ . Remark 2 :1. Observe that π = π [ N ] \ q (cid:91) i =1 π ( A i ) . 2. For each r , the ε r ( i, j ) are the the entries in the strict upper-triangle ofthe symmetric matrix M rr .3. We have δ ( z, M ∗∗ ) = v ( z, π [ N ] , ε ) and δ ( Z, M ∗ ) = q (cid:89) r =1 v ( z, π ( A r ) , ε r ) . 4. We have δ ( z, M ) = δ ( z, M ∗ ) · δ ( z, M ∗∗ ).5. For each r , we have δ r ( M ∗ ) = (cid:88) σ ∈ G r σ ( v ( z, π ( A r ) , ε r )) . 6. The ε ( i, j ) are the entries in the strict upper-triangle of the symmetricmatrix M ∗∗ . Theorem 1 : Let the notation be as above. Assume q ≥ (1) For 1 ≤ r < s ≤ q , the matrix M rs has only positive entries. (2) For 1 ≤ r < s ≤ q , the positive integer b ( m r , m s ) := (cid:107) M rs (cid:107) dependsonly on the ordered pair ( m r , m s ) and furthermore, if m r = m s , then b ( m r , m s ) is an even integer. (3) Characteristic of k is 0 and for 1 ≤ r < s ≤ q , (cid:107) M rs (cid:107) is even.7lso, assume that the properties (i) - (iv) listed below are satisfied. (i) Either m i < m j for 1 ≤ i < j ≤ q or M ∗ = 0. (ii) If properties (1) and (2) hold, then (cid:81) qr =1 δ r ( M ∗ ) (cid:54) = 0. (iii) If property (2) does not hold but properties (1) and (3) hold, then eachentry of M ∗ is an even integer. (iv) The least nonzero entry of the matrix M ∗∗ is strictly greater than thegreatest entry of the matrix M ∗ .Then Symm N ( δ ( z, M )) (cid:54) = 0. Proof : Define m = 0. At the outset, observe that a permutation σ ∈ S N canbe naturally viewed as a permutation of π [ N ] by letting σ ( i, j ) := { σ ( i ) , σ ( j ) } , i.e. , for ( i, j ) ∈ π [ N ], σ ( i, j ) := (cid:40) ( σ ( i ) , σ ( j )) if σ ( i ) < σ ( j ),( σ ( j ) , σ ( i )) if σ ( j ) < σ ( i ).Thus S N is regarded as a subgroup of the group of permutations of π [ N ].For σ ∈ S N and 1 ≤ r ≤ q , define B r ( σ ) := σ − ( A r ) = { i | ≤ i ≤ N and σ ( i ) ∈ A r } . Clearly, sets B ( σ ) , . . . , B q ( σ ) partition { , . . . , N } and B i has cardinality m i for all 1 ≤ i ≤ q .Define G := { σ ∈ S N | σ ( i, j ) ∈ π for all ( i, j ) ∈ π } . For 1 ≤ r ≤ q , a permutation σ ∈ G r is to be regarded as an element of S N bydeclaring σ ( i ) = i if i ∈ { , . . . , N } \ A r . This way each G r is identified as asubgroup of S N .Given σ ∈ G and ( i, j ) ∈ π ( A r ) with 1 ≤ r ≤ q , clearly there is a unique s with 1 ≤ s ≤ q such that σ ( i, j ) ∈ π ( A s ). Fix a σ ∈ G . Consider i ∈ B r ( σ ) ∩ A s with 1 ≤ s ≤ q . Then for i (cid:54) = j ∈ A s , we must have { σ ( i ) , σ ( j ) } in π ( A r ) andhence j ∈ B r ( σ ). It follows that A s ⊆ B r ( σ ). If 1 ≤ s < p ≤ q are such that A s ∪ A p ⊆ B r ( σ ), then an ( i, j ) ∈ A s × A p is in π whereas σ ( i, j ) is in π ( A r ).This is impossible since σ ∈ G . Thus we have established the following: given r with 1 ≤ r ≤ q and σ ∈ G , there is a unique integer r ( σ ) such that 1 ≤ r ( σ ) ≤ q and B r ( σ ) = A r ( σ ) . In other words, the image sets σ ( A ) , . . . , σ ( A q ) form apermutation of the sets A , . . . , A q . If 1 ≤ r < s ≤ q and σ ∈ G , then since r ( σ ) (cid:54) = s ( σ ), we infer that π ∩ (cid:0) A r ( σ ) × A s ( σ ) (cid:1) (cid:54) = ∅ if and only if r ( σ ) < s ( σ ).Moreover, m r ( σ ) = m r for all 1 ≤ r ≤ q and σ ∈ G .8f the first case of (i) holds, i.e. , the integers m i are mutually unequal, then wemust have r ( σ ) = r for all 1 ≤ r ≤ q and σ ∈ G . Hence, in this case G is thedirect product of (the mutually commuting) subgroups G , G , . . . , G q .Hypothesis (1) implies v ( z, π [ N ] , ε ) = v ( z, π, ε ). If G = G × G × · · · × G q ,then we have (cid:88) σ ∈ G (cid:32) q (cid:89) r =1 σ ( v ( z, π ( A r ) , ε r )) (cid:33) = q (cid:89) r =1 (cid:32) (cid:88) θ ∈ G r θ ( v ( z, π ( A r ) , ε r )) (cid:33) . For 1 ≤ r ≤ q , define w r := (cid:88) ( i,j ) ∈ π ( A r ) ε r ( i, j ) and w := q (cid:88) i =1 w i . Our hypothesis (i) assures that if m i = m j for some i (cid:54) = j , then w = 0.Now let t, t , . . . , t q , x , . . . , x N be indeterminates and let α : k [ z , . . . , z N ] → k [ t, t , · · · , t q , x , . . . , x N ]be the injective k -homomorphism of rings defined by α ( z i ) := tx i + t r if i ∈ A r with 1 ≤ r ≤ q .Then given σ ∈ S N , ( i, j ) ∈ π [ N ] and 1 ≤ r, s ≤ q , we have α ( z σ ( i ) − z σ ( j ) ) = t ( x σ ( i ) − x σ ( j ) ) + ( t r − t s )if and only if ( σ ( i ) , σ ( j )) ∈ A r × A s .Let x stand for ( x , . . . , x N ) and T stand for ( t , . . . , t q ). Given f ∈ k [ t, T, X ],by the x -degree (resp. T -degree ) of f , we mean the total degree of f in the in-determinates x , . . . , x N (resp. t , . . . , t q ). Now fix a σ ∈ G and consider V σ ( x, t, T ) := α ( σ ( v ( z, π, ε ))) . For an ordered pair ( i, j ) with 1 ≤ i, j ≤ q , set A ( σ, i, j ) := π ∩ ( A i ( σ ) × A j ( σ ) ) . It is straightforward to verify that V σ ( x, , T ) is (cid:89) ≤ r Symm ( δ ( z, E )) (cid:54) = 0, Symm ( δ ( z, E )) = 0 and Symm ( δ ( z, E )) (cid:54) = 0. Of course, in the caseof E , Theorem 1 does apply. Since (cid:107) C (cid:107) = 29 = (cid:107) C (cid:107) is an odd integer,Theorem 1 can not be applied in the case of E , E .2. For j = 1 , 2, let E j ∈ E (5 , 18) be presented in 2 × E j := (cid:20) A j A Tj B (cid:21) , where B := ,A := (cid:20) (cid:21) and A := (cid:20) (cid:21) . Then a MAPLE computation shows that h j := Symm ( δ ( z, E j )) (cid:54) = 0 for j = 1 , 2. Up to a nonzero integer multiple, h and h are the same; eitherone can be identified as the Hermite’s invariant of a quintic binary form(see [2] or [3]). Since this invariant has weight 45, it is a skew invariant.Let M ∈ E (9 , 90) be the 2 × M ij ] such that M = 0, M is the 4 × M ∈ { E , E } . Notethat Theorem 1 is applicable and thus g := Symm ( δ ( z, M )) is a nonzeroinvariant of a binary nonic. Also, since g has weight 405, g is a skewinvariant.3. Let M ∈ E (4 , 2) be the 2 × M ij ], where M = 2 D = M and M = 0 = M . Let g := Symm ( δ ( z, M )) and h := Symm ( δ ( z, M )).Then 2 M ∈ E (4 , 4) and by Lemma 1, gh (cid:54) = 0. Clearly, g and h both areinvariants of a binary quartic. A computation employing MAPLE showsthat g and h are algebraically independent over k . Lemma 2 : Suppose d is a positive integer such that N d is an integer multiple of4. Then there is an explicitly described E ∈ E ( N, d ) such that each entry of E isan even integer. Moreover, if k has characteristic 0, then g := Symm N ( δ ( z, E ))is a nonzero invariant (of degree d ) of a binary form of degree N .14 roof : First, suppose N = 2 m for some positive integer m and d is an evenpositive integer. Let E ∈ E ( N ) be the m × m block matrix [ M ij ] such that M rr := dD for 1 ≤ r ≤ m and M ij = 0 for 1 ≤ i < j ≤ m . Then clearly E ∈ E ( N, d ) and since d is even, each entry of E is an even integer. Secondly,suppose N is odd and d = 4 e for some positive integer e . Our constructionproceeds by induction on N . If N = 3, then let E := (2 e ) D . Henceforth,assume N ≥ 5. If N − M ∈ E ( N − , d ) such that each entry of M is an even integer. If N − M ∈ E ( N − , d ) such that eachentry of M is an even integer. Now let E be the 2 × C ij ] with C := (2 e ) D , C := M and C = 0 = C . Then clearly E ∈ E ( N, d ) andeach entry of E is an even integer. In either case, provided char k = 0, Lemma1 assures that g (cid:54) = 0. (cid:3) Theorem 2 : Assume that N ≥ (i) Suppose m , n are positive integers such that n ≥ N = mn . Let a , b be positive integers and let d := 2 a ( n − 1) + ( m − n − b . Then thereis an explicitly described E ∈ E ( N, d ) such that g := Symm N ( δ ( z, E )) isa (degree d ) nonzero invariant of a binary form of degree N . (ii) Suppose m , n , r are positive integers such that n ≥ 2, 1 ≤ r ≤ mn − N = 2 mn − r . Given positive integers a , b such that c := 2( n − a + ( m − n − br is an integer,there is an explicitly described E ∈ E ( N, mnc ) yielding a (degree mnc )nonzero invariant g := Symm N ( δ ( z, E )) of a binary form of degree N . (iii) Suppose l , m , n are positive integers such that l < m < n < l + m and N = l + m + n . Given a positive integer d such that each of a := ( m + l − n ) d lm , b := ( l + n − m ) d ln , c := ( m + n − l ) d mn is an integer, there is an explicitly described E ∈ E ( N, d ) yielding a (degree d ) nonzero invariant g := Symm N ( δ ( z, E )) of a binary form of degree N . (iv) Suppose s is a nonnegative integer and t , u , v are positive integers suchthat t ≤ u ≤ t − 1. Then letting N := 2(2 tv + 1) and d := (2 s + 1)(2 u + 1)(4 uv + 2 v + 1) , there is an explicitly described E ∈ E ( N, d ) such that g := Symm N ( δ ( z, E ))is a nonzero invariant of a binary form of degree N . Moreover, g is a skewinvariant of weight w := (2 s + 1)(2 tv + 1)(2 u + 1)(4 uv + 2 v + 1). (v) Given E ∈ E ( N, d ) such that each entry of E is strictly less than d and Symm N ( δ ( z, E )) (cid:54) = 0, a matrix E ∗ ∈ E (2 N − , dN ) can be so constructed15hat g := Symm N ( δ ( z, E ∗ )) is a nonzero invariant of a binary form of de-gree 2 N − Proof : To prove (i), let E ∈ E ( N ) be the n × n block matrix [ M ij ], where M ii = 0 for 1 ≤ i ≤ n and M ij = 2 aI + bD m for 1 ≤ i < j ≤ n . It isstraightforward to verify that E ∈ E ( N, d ) and Theorem 1 can be applied todeduce g (cid:54) = 0.To prove (ii), first note that mn − r ≥ 1. Let E ∈ E ( N ) be the ( n +1) × ( n +1)block matrix [ M ij ] defined as follows. For 1 ≤ i ≤ n +1, M ii = 0. If mn − r ≤ m ,then for 1 ≤ i < j ≤ n + 1, M j is the ( mn − r ) × m matrix having each entryequal to c and M ij = 2 aI + bD m . If m < mn − r , then for 1 ≤ i < j ≤ n + 1, M ij = 2 aI + bD m and M i ( n +1) is the m × ( mn − r ) matrix having each entryequal to c . Then clearly E ∈ E ( N, d ). If mn − r = m , then m ( mn − r ) c =2 ma + m ( m − b is necessarily an even integer. Now it is straightforward toverify that Theorem 1 can be employed to infer g (cid:54) = 0.To prove (iii), let E ∈ E ( N ) be the 3 × M ij ] such that M rr = 0 for 1 ≤ r ≤ M = M T is the l × m matrix having each entryequal to a , M = M T is the l × n matrix having each entry equal to b and M = M T is the m × n matrix having each entry equal to c . By hypothesis,each of a , b , c is a positive integer. Since d = ma + nb = la + nc = lb + mc ,we have E ∈ E ( N, d ). As before, it is easily verified that Theorem 1 is indeedapplicable in this case and hence g (cid:54) = 0.To prove (iv), let m := 1, n := 4 uv + 2 v + 1 and r := 8 uv − tv + 4 v . Clearly, n ≥ N = 2 mn − r . Since t ≤ u ≤ t − 1, we have 1 ≤ r ≤ n − 1. Define a := (2 s + 1)(2 u − t + 1) and say b := 1. Then letting c := (2 s + 1)(2 u + 1), wehave c ≥ cr = ( n − a + ( m − b ]. Observe that the positive integers a , b , c , m , n , r satisfy all the requirements of (ii). Thus, by taking E ∈ E ( N, d )as described in the proof of (ii), we infer that g (cid:54) = 0. If w denotes the weight of g , then 2 w = N d and hence w = (2 s + 1)(2 tv + 1)(2 u + 1)(4 uv + 2 v + 1). Since w is an odd integer, g is a skew invariant.Lastly, to prove(v), suppose E ∈ E ( N, d ) is such that each entry of E isstrictly less than d and Symm N ( δ ( z, E )) (cid:54) = 0. Let E ∗ be the 2 × C ij ], where C := 0, C := E and C = C T is the ( N − × N matrix witheach entry equal to d . Clearly, E ∗ ∈ E (2 N − , dN ) and Theorem 1 can beapplied to infer g (cid:54) = 0. (cid:3) Example 2 : We continue assuming N ≥ N = 4 e . Using (i) of Theorem 2 with n := 2 and m := 2 e , we obtainnonzero invariants of degree d for d = 2 e + 1 and all d ≥ N − 1. If char k = 0 and d ≤ N − d .2. With the notation of (iii), let Y := { ≤ d ∈ Z | a, b, c ∈ Z } and y := 2 lmngcd ( N − l, N − m, N − n, lmn ) . d ∈ Y if and only if d = sy forsome positive integer s . Of course, 2 lmn ∈ Y ; but y can be strictly lessthan 2 lmn ( e.g. , consider ( l, m, n ) := (2 , , 6) or ( l, m, n ) := (9 , , l + m + n is odd and d = 2 mod 4, then the resulting g is a nonzeroskew invariant. So, (iii) produces skew invariants for binary forms of odddegrees (in contrast to (iv)). The least value of N for which (iii) may beused to obtain skew invariants, is N = 3 + 5 + 7 = 15; whereas for theones that can be obtained by using (iv) is N = 2(2 · · N = l + m + n with l ≤ m ≤ n < l + m , by imposingadditional requirements such as: ( l + m − n ) d is divisible by 4 if l = m and so on, hypotheses of Theorem 1 can be satisfied. Assertion (iii) canbe generalized for certain types of partitions of N into 4 or more parts;the task of formulating such generalizations is left to the reader.3. Let E ∈ { E , E } ⊂ E (5 , E , E are as in the second exampleabove Theorem 2. For 2 ≤ n ∈ Z , let d n , M n ∈ E (2 n + 1 , d n ) be induc-tively defined by setting d := 18, M := E , d n +1 := (2 n + 1) d n and where M n +1 := M ∗ n , is derived from M n as in (iv) of Theorem 2. Then by (v) ofTheorem 2, g n := Symm n +1 ( δ ( z, M n )) is a nonzero skew invariant of abinary form of degree 2 n + 1 for 2 ≤ n ∈ Z . Remark 4 : Theorem 2 exhibits the simplest applications of Theorem 1. Atpresent, there does not exist a characterization of pairs ( N, d ) for which Theo-rem 1 can be used to obtain a nonzero invariant. Interestingly, it is impossibleto use Theorem 1 to construct invariants corresponding to certain pairs ( N, d ), e.g , consider ( N, d ) = (5 , Definitions : Let n , s be a positive integers.1. Let (cid:22) denote the lexicographic order on Z s +1 .2. For α := ( a , . . . , a s +1 ) ∈ Z s +1 , let | α | := (cid:80) s +1 i =1 a i and wt ( n, α ) := 12 (cid:34) n − (cid:32) s +1 (cid:88) i =1 a i (cid:33)(cid:35) . 3. Define ℘ ( s, n ) := ( ℘ ( s, n ) , . . . , ℘ s +1 ( s, n )) ∈ Z s +1 , where ℘ j ( s, n ) := (cid:22) n − (cid:80) ≤ i ≤ j − ℘ i s + 2 − j − ( s + 1 − j )2 (cid:23) for 1 ≤ j ≤ s + 1.4. Let (cid:36) ( s, n ) := wt ( n, ℘ ( s, n )). 17. By (cid:61) ( s, n ) we denote the set of all α := ( a , . . . , a s +1 ) ∈ Z s +1 such that a < a < · · · < a s +1 and | α | = n . Let P ( s, n ) be the subset of (cid:61) ( s, n )consisting of ( a , . . . , a s +1 ) ∈ (cid:61) ( s, n ) with a ≥ i, j ) ∈ Z with 1 ≤ i < j ≤ s + 1, let η ( i, j ) := ( η , . . . , η s +1 ) where η r = 0 if r (cid:54) = i, j , η i = 1 and η j = − 1. An ( s + 1)-tuple β is said to bean elementary modification of α ∈ Z s +1 provided β = α + η ( i, j ) for some1 ≤ i < j ≤ s + 1. An ( s + 1)-tuple β is said to be a modification of α ∈ Z s +1 if there is a finite sequence α = α , . . . , α r = β such that α i isan elementary modification of α i − for 2 ≤ i ≤ r . Lemma 3 : Fix positive integers n , s and let e be the integer such that n − s ( s + 1)2 = (cid:22) ns + 1 − s (cid:23) ( s + 1) + e. Let ℘ ( s, n ) = ( p , . . . , p s +1 ). Then, the following holds. (i) We have p j = (cid:26) p + j − ≤ j ≤ s + 1 − e , and p + j if s + 2 − e ≤ j ≤ s + 1.In particular, ℘ ( s, n ) ∈ (cid:61) ( s, n ). Moreover, if ( s + 1)( s + 2) ≤ n , then ℘ ( s, n ) ∈ P ( s, n ). (ii) We have (cid:36) ( s, n ) = ( s + 1)( s + 2)2 (cid:22) ns + 1 − s (cid:23) + ( s + 1) ( s + 2) − n ( s + 2)2 (cid:22) ns + 1 − s (cid:23) + 3( s + 1) + 2( s + 1) − n )( s + 1) − n )( s + 1) + 24 n . (iii) Let α := ( a , . . . , a s +1 ) ∈ (cid:61) ( s, n ). Then, α (cid:22) ℘ ( s, n ), ℘ ( s, n ) is a modifi-cation of α and (cid:88) ≤ i 2, ( s + 1)( s + 2) ≤ n and p + e = bs + d where b , d are nonnegative integers with d ≤ s − 1. Then, letting ℘ ( s − , n ) :=( q , . . . , q s ), we have q = p + b + 1 and (cid:36) ( s, n ) − (cid:36) ( s − , n ) = p ( s + 1 − e ) + bd ( s + 1) + 12 b ( b − s ( s + 1) . In particular, q > p and (cid:36) ( s, n ) − (cid:36) ( s − , n ) ≥ p . If p = 1, then2 ≤ q ≤ ≤ (cid:36) ( s, n ) − (cid:36) ( s − , n ) ≤ s + 2. (vi) Suppose s ≥ 2, ( s + 1)( s + 2) ≤ n and let v ( s, n ) := ( v , . . . , v s ) where v i := i for 1 ≤ i ≤ s and v s = n − (1 / s ( s + 1). Then, v ( s, n ) (cid:22) α and wt ( n, v ( s, n )) ≤ wt ( n, α ) for α ∈ P ( s, n ). Proof : Note that 0 ≤ e ≤ s and hence s + 1 − e ≥ 1. Suppose 1 ≤ j ≤ s + 1 − e is such that p i = p + i − ≤ i ≤ j . Then, p j +1 = (cid:22) p − j ( j − − s ( s + 1) − e + ( s − j )( s + 1 − j )2( s + 1 − j ) (cid:23) = (cid:22) p + j + es + 1 − j (cid:23) . If j < s + 1 − e , then e < s + 1 − j and hence p j +1 = p + j . If j = s + 1 − e , then p j +1 = p + j + 1. Next suppose (i) holds for some j with s + 2 − e ≤ j ≤ s .Then, p j +1 = (cid:22) p − j ( j − − s ( s + 1) + 2( j + e − s − − e + ( s − j )( s + 1 − j )2( s + 1 − j ) (cid:23) = p + j + 1 . Clearly, p < p < · · · < p s +1 and if ( s + 1)( s + 2) ≤ n , then p ≥ 1. Also, | ℘ ( s, n ) | = p ( s + 1) + [ s ( s + 1) / 2] + e = n . Thus (i) holds.Let u ( X ) , v ( X ) ∈ Z [ X ] be defined by v ( X ) = s +1 (cid:89) j =0 ( X + p + j ) = ( X + p + s + 1 − e ) u ( X ) . Then, (cid:36) ( s, n ) is the coefficient of X s − in u ( X ). The coefficient of X s in v ( X − p ) is 12 (cid:32) s +1 (cid:88) i =0 i (cid:33) − s +1 (cid:88) i =0 i = (3 s + 5)( s + 2)( s + 1) s 24. Now a straightforward computation verifies (ii).Obviously, wt ( n, α ) < n for all α ∈ (cid:61) ( s, n ). If β ∈ (cid:61) ( s, n ) is an elementarymodification of α = ( a , . . . , a s +1 ) ∈ (cid:61) ( s, n ), then note that wt ( n, β ) > wt ( n, α ).19ence α has a modification v ∈ (cid:61) ( s, n ) that is ‘final’ in the sense that no memberof (cid:61) ( s, n ) is an elementary modification of v . Fix such v := ( v , . . . , v s +1 ). If1 ≤ i ≤ s + 1 is such that v i +1 > v i + 2, then v + η ( i, i + 1) ∈ (cid:61) ( s, n ); thiscontradicts our assumption about v . So, v i + 1 ≤ v i +1 ≤ v i + 2 for all 1 ≤ i ≤ s .If there are 1 ≤ i < j ≤ s +1 such that v i +1 = v i +2 as well as v j +1 = v j +2, then v + η ( i, j ) ∈ (cid:61) ( s, n ); an impossibility. Hence a i +1 = a i + 2 for at most one i with1 ≤ i ≤ s . Consequently, n = | v | = ( s +1) v +( s +1 − j )+[ s ( s +1) / 2] for some j with 1 ≤ j ≤ s +1. Clearly, j = s +1 − e and in view of (ii), we have v = ℘ ( s, n ).Thus ℘ ( s, n ) is a modification of α . In particular, wt ( n, α ) ≤ (cid:36) ( s, n ) and α (cid:22) ℘ ( s, n ). The equality displayed on the left in (iii) readily follows from thedefinition of wt ( n, α ). Thus (iii) holds.Assertion (iv) is simple to verify. To prove (v), assume s ≥ p + e = bs + d where b , d are nonnegative integers with d ≤ s − 1. Consequently, q = p + b + 1 > p . Using (ii) (cid:36) ( s, n ) − (cid:36) ( s − , n ) can be computed in astraightforward manner. If e ≤ s − 1, then (cid:36) ( s, n ) − (cid:36) ( s − , n ) is clearly ≥ p .If e = s , then we have b ≥ b − s = p − d , (cid:36) ( s, n ) − (cid:36) ( s − , n ) ≥ p (cid:18) b ( s + 1) (cid:19) ≥ p . If p = 1, then since 0 ≤ e ≤ s and s ≥ 2, we have 0 ≤ b ≤ 1. If e ≤ s − 2, then b = 0 and hence q = 2, (cid:36) ( s, n ) − (cid:36) ( s − , n ) = s + 1 − e ≤ s + 1. If e = s − b = 1, d = 0 and hence q = 3, (cid:36) ( s, n ) − (cid:36) ( s − , n ) = 2. Lastly, if e = s ,then b = 1 = d and hence q = 3, (cid:36) ( s, n ) − (cid:36) ( s − , n ) = s + 2. This establishes(v). The proof of (vi) is left to the reader. (cid:3) Lemma 4 : Let m, n, t ∈ Z and ( b , . . . , b m ) ∈ Z m be such that m ≥ n ≥ b + · · · + b m = t and b i ≥ ≤ i ≤ m . Let t = qn + r , where q , r areintegers with q ≥ ≤ r < n . Then, there exists an m × n matrix A := [ a ij ]satisfying the following. (i) ≤ a ij ∈ Z for 1 ≤ i ≤ m , 1 ≤ j ≤ n and (cid:107) A (cid:107) = t . (ii) c j ( A ) := r j (cid:0) A T (cid:1) = (cid:26) q + 1 if 1 ≤ j ≤ r and q if r + 1 ≤ j ≤ n . (iii) r i ( A ) = b i for 1 ≤ i ≤ m . Proof : Let t = qn + r , where q , r are integers with q ≥ ≤ r < n . Ourproof proceeds by induction on m . If m = 1, then let a j := q + 1 if 1 ≤ j ≤ r and a j := q if r + 1 ≤ j ≤ n . Henceforth suppose m ≥ b m = (cid:96)n + ρ where (cid:96) , ρ are integers with (cid:96) ≥ ≤ ρ < n .Case 1: ρ ≤ r . By our induction hypothesis there is an ( m − × n matrix[ a ij ] such that 0 ≤ a ij ∈ Z for 1 ≤ i ≤ m − ≤ j ≤ n , (cid:107) A (cid:107) = t − b m , a j + · · · + a ( m − j = q − (cid:96) + 1 for 1 ≤ j ≤ r − ρ , a j + · · · + a ( m − j = q − (cid:96) for r − ρ + 1 ≤ j ≤ n and a i + · · · + a in = b i for 1 ≤ i ≤ m − 1. Define a mj := (cid:96) for20 ≤ j ≤ r − ρ , a mj := (cid:96) + 1 for r − ρ + 1 ≤ j ≤ r and a mj := (cid:96) for r + 1 ≤ j ≤ n .Then, the resulting m × n matrix [ a ij ] is clearly the desired matrix A .Case 2: ρ > r . At the outset observe that r < n + r − ρ < n . As before,our induction hypothesis assures the existence of an ( m − × n matrix [ a ij ]such that 0 ≤ a ij ∈ Z for 1 ≤ i ≤ m − ≤ j ≤ n , (cid:107) A (cid:107) = t − b m , a j + · · · + a ( m − j = q − (cid:96) for 1 ≤ j ≤ n + r − ρ , a j + · · · + a ( m − j = q − (cid:96) − n + r − ρ + 1 ≤ j ≤ n and a i + · · · + a in = b i for 1 ≤ i ≤ m − 1. Define a mj := (cid:96) + 1 for 1 ≤ j ≤ r , a mj := (cid:96) for r + 1 ≤ j ≤ n + r − ρ and a mj := (cid:96) + 1for n + r − ρ + 1 ≤ j ≤ n . Then, the resulting m × n matrix [ a ij ] is the desiredmatrix A . (cid:3) Definitions : Let n and w be positive integers.1. Define β ( n ) := (cid:22) √ n + 1 − (cid:23) . 2. For an integer s with 1 ≤ s ≤ β ( n ) − a := ( m , . . . , m s +1 ) ∈ P ( s, n ), define ν ( w, a ) := (cid:18) s − w − wt ( n, a ) s − (cid:19) and d ( w, a ) := n − w − wt ( n, a ) if m = 1, n − w − wt ( n, a ) if w = 1 + wt ( n, a ), n − m + 1 + (cid:108) w − wt ( n, a ) m (cid:109) otherwise.3. Let ν ( w, s, n ) := ν ( w, ℘ ( s, n )) and d ( w, s, n ) := d ( w, ℘ ( s, n )). Theorem 3 : Assume that N is an integer ≥ k is a field of characteristiceither 0 or strictly greater than N . Let F be the generic binary form of degree N (as in the introduction). Let s be an ineteger with 1 ≤ s ≤ β ( N ) − a := ( m , . . . , m s +1 ) ∈ P ( s, N ). Let m := m and let w be an integer suchthat θ := w − wt ( N, a ) ≥ 1. Then, for a positive integer d ≥ d ( w, a ), there ex-ist ν ( w, a ) k -linearly independent semi-invariants of F of weight w and degree d . Proof : Fix an ordered s -tuple ( θ , . . . , θ s ) of nonnegative integers with θ + · · · + θ s = θ. Since θ ≥ 1, using Lemma 4 we obtain an s × m matrix B ∗ := [ b ∗ ij ] havingnonnegative integer entries such that r i ( B ∗ ) = θ i for 1 ≤ i ≤ s and (cid:98) θ/m (cid:99) ≤ c m ( B ∗ ) ≤ · · · ≤ c ( B ∗ ) = (cid:100) θ/m (cid:101) . u be the greatest positive integer such that c u ( B ∗ ) ≥ v be theleast positive integer with b ∗ vu ≥ 1. Define an s × m matrix B := [ b ij ] as follows.If u = 1 (in particular, if m = 1), let B = B ∗ . If u ≥ 2, then let b ij := b ∗ ij for ( i, j ) (cid:54) = ( v, , ( v, u ), let b vu := b ∗ vu − b v := b ∗ v + 1. Then, B hasnonnegative integer entries, r i ( B ) = θ i for 1 ≤ i ≤ s , c ( B ) = min { (cid:100) θ/m (cid:101) , θ } , and (cid:98) θ/m (cid:99) − ≤ c j ( B ) ≤ (cid:100) θ/m (cid:101) , for 2 ≤ j ≤ m .Using Lemma 4 again, we obtain matrices A , . . . , A s with nonnegative integerentries such that(1) A l has size m × m l +1 for 1 ≤ l ≤ s ,(2) r i ( A l ) = b li for 1 ≤ l ≤ s , 1 ≤ i ≤ m and(3) (cid:98) θ l /m (cid:99) ≤ c j ( A l ) ≤ c j − ( A l ) ≤ (cid:100) θ l /m (cid:101) for 2 ≤ j ≤ m l +1 .Clearly, (cid:107) A l (cid:107) = θ l for 1 ≤ l ≤ s . Furthermore, we have(4) r ( A ) + · · · + r ( A s ) = min { (cid:100) θ/m (cid:101) , θ } , and(5) r i ( A ) + · · · + r i ( A s ) ≤ (cid:100) θ/m (cid:101) for 2 ≤ i ≤ m .Let I denote a matrix (of any chosen size) having each entry 1. Let M := [ M ij ]be an ( s + 1) × ( s + 1) block-matrix such that M ji is the transpose of M ij for1 ≤ i ≤ j ≤ s + 1, and the block M ij is a m i × m j matrix defined by M ij := i = j , I + A j − if i = 1 < j ≤ s + 1, I if 2 ≤ i < j ≤ s + 1.Let M (cid:48) denote the ( N − × ( N − 1) matrix obtained from M by deleting thefirst row as well as the first column of M . Then, M ∈ E ( N ) and M (cid:48) ∈ E ( N − r ( M ) = d ( w, a ) > r i ( M ) for 2 ≤ i ≤ N ,and each of M , M (cid:48) satisfies requirements (1), (2), (i) - (iv) of Theorem 1.Hence letting φ ( θ , . . . , θ s ) := Symm N ( δ ( z, M )), we have φ ( θ , . . . , θ s ) (cid:54) = 0 aswell as Symm N − ( δ ( z, M (cid:48) )) (cid:54) = 0. Observe that the coefficient of z d ( w,α )1 in φ ( θ , . . . , θ s ) is the symmetrization of δ ( z (cid:48) , M (cid:48) ) where z (cid:48) := ( z , . . . , z N ). Since Symm N − ( δ ( z, M (cid:48) )) (cid:54) = 0, we conclude that the z -degree (and hence also each z i -degree) of φ ( θ , . . . , θ s ) is exactly d ( w, a ). Let α be the k -monomorphismemployed in Theorem 1. Then, as noted in no. 2 of Remarks 3, the t -initialcoefficient of α ( φ ( θ , . . . , θ s )) is a nonzero constant ( i.e. , element of k ) multipleof η ( θ , . . . , θ s ) := (cid:89) ≤ j ≤ s ( t − t j +1 ) θ j (cid:89) ≤ i 7. It is essential to pointout that the lower bounds proved in [4], [12], [19] assume N ≥ 8. To the bestof our knowledge, there is nothing in the existing literature with which we cancompare the bounds in examples below.1. If N = 3, then s = 1 and (cid:36) (1 , 3) = 2. In this case, Theorem 3 implies thatfor 0 ≤ n ∈ Z , there exists a nonzero semi-invariant (of a binary cubicform F ) of weight 2 + n and degree at least 2 + n .2. If N = 4, then s = 1 and (cid:36) (1 , 4) = 3. In this case, Theorem 3 implies thatfor 0 ≤ n ∈ Z , there exists a nonzero semi-invariant (of a binary quarticform F ) of weight 3 + n and degree at least 3 + n .3. If N = 5, then s = 1 and (cid:36) (1 , 5) = 6. In this case, Theorem 3 implies thatfor 0 ≤ n ∈ Z , there exists a nonzero semi-invariant (of a binary quinticform F ) of weight 6 + n and degree at least 4 + (cid:100) n/ (cid:101) . Note that for thepartition 1 < 4, we can use Theorem 1 to verify the existence of a nonzerosemi-invariant of weight 4 + n and degree at least 4 + n . So, we obtaintwo k -linearly independent semi-invariants of weight 6 + n and degree atleast 6 + n .4. Assume N = 6. Then 1 ≤ s ≤ (cid:36) (1 , 6) = 8 and (cid:36) (2 , 6) = 11. Taking s = 1 in Theorem 3, we infer the existence of a nonzero semi-invariant (ofa binary sextic form F ) of weight 8 + n and degree at least 8 + n for all0 ≤ n ∈ Z . Next, taking s = 2, Theorem 3 assures the existence of 5 + nk -linearly independent semi-invariants of weight 16 + n and degree at least10 + n for all 0 ≤ n ∈ Z .5. Assume N = 7. Then 1 ≤ s ≤ (cid:36) (1 , 7) = 12 and (cid:36) (2 , 7) = 14. Letting s = 1 in Theorem 3, we obtain a nonzero semi-invariant (of a binary hepticform F ) of weight 12 + n and degree at least 5 + (cid:100) n/ (cid:101) for 0 ≤ n ∈ Z .Using Theorem 1 for the partition 2 < 5, we infer the existence of anonzero semi-invariant of weight 10 + n and degree at least 6 + (cid:100) n/ (cid:101) forall 0 ≤ n ∈ Z . Letting s = 2 in Theorem 3, we deduce the existence of5 + n k -linearly independent semi-invariants of weight 18 + n and degreeat least 5 + (cid:100) ( n + 4) / (cid:101) for all 0 ≤ n ∈ Z . Remark 5 : Let N , w and d are positive integers. Let P P ( N, w, d ) := (cid:24) · (min { w, d , N } ) − · √ min { w, d N } (cid:25) . 23f min { N, d } ≥ w ≤ N d/ 2, then by Theorem 1.2 of [12], there are at least P P ( N, w, d ) k -linearly independent semi-invariants (of a binary N -ic form F )of degree d and weight w . Observe that for ( w, d ) with w ≥ N / d ≥ N ,the bound P P ( N, w, d ) is independent of ( w, d ) ( i.e. , depends only on N ). Incontrast, the lower bound ν ( w, a ) is a polynomial of degree s − w . Thereader may wish to make similar comparison with results of [4]. Example 4 : Let ν ( w, N ) := ν ( w, β ( N ) − , N ). Consider the case of N = 15.Note that β ( N ) = 5 and P (4 , 15) = { ℘ (4 , } . We have (cid:36) (4 , 15) = 85 and ℘ (4 , 15) = 1. Let ν ( w ) := ν ( w, , ≤ n ∈ Z , we have at least ν (85 + n ) k -linearly independent semi-invariants ofweight 85 + n and degree d ≥ 14 + n . Observe that 2(85 + n ) < (14 + n ) for n ≥ N = 225 < n ) for n ≥ 28 and ν (85 + n ) = (cid:18) n (cid:19) = 16 n + n + 116 n + 1 for n ≥ P P (15 , 85 + n, d ) = 1 < ν (85 + n )for all n ≥ d ≥ 14 + n . Let semdim ( w, d, N ) denote the dimension of the k -vector space of semi-invariants (of our N -ic form F ) of weight w and degree d . Assume k has characteristic 0. Then, in the notation of the introduction, semdim ( w, d, N ) is p w ( N, d ) − p w − ( N, d ) := the coefficient of q w in ( q − (cid:18) N + dd (cid:19) q . The table below presents a MAPLE computation of ν (85 + n ) and semdim (85 + n, 14 + n, 15) (denoted by semdim ) for a small sample of values of the weight w ( i.e. , values of n ). w ν ( w ) semdim 95 286 1020697105 1771 4232793115 5456 11374824 w ν ( w ) semdim 125 12341 25995316135 23426 54621331145 39711 108639772Let s = 3 and a := v (3 , 15) = (1 , , , n ≥ 0, wehave ν (65 + n, a ) = (1 / n + 2)( n + 1) and d (65 + n, a ) = 14 + n . 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