Some properties of hypergeometric series associated with mirror symmetry
|
|
- Amos Harper
- 5 years ago
- Views:
Transcription
1 Some properties of hypergeometric series associated with mirror symmetry Don Zagier and Aleksey Zinger March 4, 2008 Abstract We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witten invariants of any Calabi-Yau projective hypersurface. 1. Introduction An astounding prediction for the genus zero Gromov-Witten invariants of counts of rational curves in) a quintic threefold was made in [CaDGP]. It was formulated in terms of the function F defined in 1) below and related objects. This 1991 mirror symmetry prediction was mathematically verified about five years later. The 1993 mirror symmetry prediction of [BCOV] for the genus one Gromov-Witten invariants of a quintic threefold was verified in [Z1], and a generalization to degree n hypersurfaces in CP n 1 for arbitrary n was formulated and proved in [Z2]. The proofs in these two papers make use of the properties of F described by Theorems 1 3 below. Theorem 4 explores related properties of F which appear to be of interest in their own right and may also be of use in computation of higher genus Gromov-Witten invariants. Some further conjectural properties are stated in Section 3. We denote by P 1 + xqw)[[x]] the subgroup of power series in x with constant term 1 whose coefficients are rational functions in w which are holomorphic at w =0. Thus, the evaluation map P 1 + xq[[x]], F w, x) F 0, x), is well-defined. We define a map M : P P by { MF w, x) = 1 + x } F w, x) w x F 0, x). Our first result says that the hypergeometric functions arising in the mirror symmetry predictions are periodic fixed points of the map M. 1
2 Theorem 1. Let n be a positive integer and F P the hypergeometric series Fw, x) = d=0 x d r=nd nw+r) r=d w+r)n w n ). 1) Then M n F = F. Note that we consider n as fixed and therefore omit it from the notations. If we now define further power series F p P and I p 1 + xq[[x]] for all p 0 by F p w, x) = M p Fw, x), I p x) = F p 0, x), so that F p+1 = 1 + w 1 x d/dx)f p /I p ), then Theorem 1 says that F n+p = F p and consequently I n+p = I p for all p 0. The next result gives further properties of the functions {I p } p Z/nZ. Theorem 2. The power series I p x), 0 p n 1, satisfy I 0 x) I 1 x) I n 1 x) = 1 n n x) 1, 2) I 0 x) n 1 I 1 x) n 2 I n 1 x) 0 = 1 n n x) n 1)/2, 3) I p x) = I n 1 p x) 0 p n 1). 4) We note that 2) and the symmetry property 4) imply 3). However, 3) is simpler to prove directly than 4) and will be verified together with 2) before we give the proof of 4). The power series I p describe the structure of F at w =0. We will also give information about its structure at w =. We begin with the following observation, which will be proved in Subsection 2.3. Lemma 1. If F P and M k F = F for some k > 0, then every coefficient of the power series log F w, x) Qw)[[x]] is Ow) as w. Applying this lemma to F = F, which satisfies its hypothesis by Theorem 1, we find that log Fw, x) has an asymptotic expansion j= 1 µ jx)w j with µ j x) xq[[x]] for all j 1 or equivalently, that Fw, x) itself has an asymptotic expansion Fw, x) e µx)w for some power series µ = µ 1, Φ 0 = e µ 0, Φ 1 = Φ 0 µ 1,... in Q[[x]]. s=0 Φ s x) w s w ) 5) Theorem 3. The first three coefficients µx), Φ 0 x), and Φ 1 x) in the expansion 5) are given by µx) = x 0 Lu) 1 u du, Φ 0 x) = Lx), Φ 1 x) = n 2)n+1) 24n where Lx) denotes the power series 1 n n x) 1/n Z[[x]]. Lx) Lx) n ), 6) 2
3 The proof of this theorem in Subsection 2.3 can be systematized and streamlined to obtain an algorithm for computing every Φ s by a differential recursion, which we now state. For integers m j 0 and for our fixed integer n) we define H m,j = H m,j X) Q[X] inductively by H 0,j = δ 0,j, H m,j = H m 1,j + X 1) X d dx + m j ) H m 1,j 1 for m 1 7) n with the convention that H m 1,j 1 = 0 if j = 0). For example, for 0 j 2 we find H m,0 X) = 1, H m,1 X) = 1 ) m X 1), n 2 H m,2 X) = 1 ) ) m n ) 3 m + 1)X 1 X 1) + n 2 3 n 2 X 1) 2 ; 4 8) more generally, H m,j for fixed j 1 and varying m has the form j m k=1 j+k) Qj,k X) with Q j,k Z[n 1, X] defined inductively by Q 0,k = δ 0,k, Q j,k = X 1) XQ j 1,k + k Q j 1,k + k + j 1) Q j 1,k 1 )/n ) for j 1. We now define differential operators L k 0 k n) on Q[[x]] by L k = k ) n k i ) ) n r H n i,k i L n ) L n Sr n) 1) i i n r H n i r,k i r L n ) D i, 9) i=0 where D = x d/dx and S r n) denotes the rth elementary symmetric function of 1, 2,..., n a Stirling number of the first kind). Using 8), we find that the first two of these operators are L 1 = nd L n 1) = nldl 1, 10) ) n L 2 = D 2 3n 1) L n 1)D + n 1 ) n 2)n 11) L n 1 L n 1). 11) 2 2 n 24 Theorem 4. i) The power series Φ s Q[[x]], s 0, are determined by the first-order ODEs L 1 Φ s ) + 1 L L 2Φ s 1 ) + 1 L 2 L 3Φ s 2 ) L n 1 L nφ s+1 n ) = 0, s 0, 12) with the convention Φ r = 0 for r < 0) together with the initial condition Φ s 0) = δ 0,s. ii) For fixed s and n, Φ s x) belongs to L Q[L]. iii) For fixed s, Φ s x) belongs to Qn)[L, L 1, L n ]. The meaning of part iii) in Theorem 4 is that for each s 0 there exists Ψ s Ψ s a, X, Y, Z) Qa)[X, Y, Z] such that the function Φ s x) defined by 1) and 5) is given by Φ s x) = Ψ s n, Lx), Lx) 1, Lx) n). In particular, iii) neither implies nor is implied by ii). 3
4 For example, from 12) for s = 0 and s = 1 together with equations 10) and 11) one finds the second and third identity in 6), and continuing the same way one obtains Φ 2 = n + 1)2 n 2) n) 2 L 2L n + L 2n 1 ) = Φ 2 / 1 2LΦ0, { n + 1)n 2) Φ 3 = 30 24n) n n n n 164) L 3n 2 72 n 1)3n 1)7n 2 9n + 14) L 2n n + 1) 2 n 2) 2 L 2n 1 L n) + 72 n 1)7n 3 17n n 24) L n 2 } + 5n n 3 447n n 556) L, illustrating parts ii) and iii) of the theorem. These expressions, and the similar formulas obtained for s 7, suggest that in fact Φ s for s fixed and n varying is an element of Q[n, n 1, L, L 1, L n ], sharpening statement iii), but we do not know how to prove this. Some further data and a further conjecture concerning the functions Φ s x) are given in Section 3. Remark. Although the calculations in this paper are completely independent of the theory of Gromov-Witten invariants, this theory does form the principal motivation, so we give here a brief indication of how the function F defined in 1) is used in GW-theory. Mathematical formulations of mirror symmetry ordinarily involve the coefficients of w 0,..., w n 1 of the Taylor series expansion of the function F 0 in equation 25) at w = 0. These coefficients are of course the same as the corresponding coefficients of F. The power series F 0 and F are the evaluations at α 1 =... = α n = 0 and = 1 of the functions Ỹ 0 Y d=0 d=0 x d x d r=d r=nd nw+r ) k=n k=1 w α k+r ) r=nd nw+r ) and r=d k=n k=1 w α k+r ) k=n k=1 w α k) ), which appear prominently in the proofs of genus 0 mirror symmetry in [Gi] and of genus 1 mirror symmetry in [Z1], respectively; both proofs involve a localization computation on P n 1. As elements of HT Pn 1 ; Q) [[ 1, x ]] Q[α 1,..., α n ] / k=n [[ w α k )) 1, x ]], where T is the n-torus acting in the usual way on P n 1, Ỹ0 and Y are the same. On the other hand, k=1 Res =0 { Yw = αi ) } = { Res =0 Yw) } w=αi. This property, which clearly does not hold for Ỹ0 viewed as a power series in rational functions), is used in an essential way in Subsection 3.4 in [Z1] and eventually reduces the problem of computing genus 1 GW-invariants of X n to determining the first three coefficients in the expansion 5) of F. 4
5 2. Proofs 2.1. Preliminaries It will be convenient to introduce notations D and D w for the first order differential operators D = x d dx and D w = D + w on Qw)[[x]]. Here we think of w as a parameter rather than a variable and write simply d dx instead of x.) The effect of D w on a power series c d w)x d Qw)[[x]] is to multiply each c d w) by w + d, so D w has an inverse operator Dw 1 which replaces each c d w) by w + d) 1 c d w). The operator M defined above can be written in terms of D w as F w, x) w 1 [ ] D w F w, x)/f 0, x). We remark that instead of working with the functions F p w, x), we could have worked with the functions R p w, t) = e wt F p w, e t ), which are the objects that actually arise in the analysis of the mirror symmetry predictions for Gromov-Witten invariants. If we had done that, then the differential operator D w = w + x d/dx would have been replaced by the simpler differential operator d/dt, explaining why this operator plays such a ubiquitous role in our analysis. But it is easier, both in the calculations and for purposes of exposition, to work with power series over Qw) in a single variable x rather than with objects in the less familiar space e wt Qw)[[e t ]]. The following lemma and its corollary are key to the proofs of the four theorems stated above. Lemma 2. Suppose c 0,..., c m, f, g, a are functions of t with f not identically 0) satisfying c m f m) + c m 1 f m 1) c 0 f = 0, c m g m) + c m 1 g m 1) c 0 g = a, 13) where f k) = d k f/dt k. Then the function h := g/f) satisfies c m 1 h m 1) + c m 2 h m 2) c 0 h = a, 14) where c s t) = m r r=s+1 s+1) cr t) f r 1 s) t). Proof: Using Leibnitz s rule and 13), we find a = m ) r) m c r f g/f = c r f r) g/f + r 1 s=0 ) r )f r 1 s) h s) s + 1 = m 1 s=0 c s h s). Corollary 1. Suppose F w, x) P satisfies m ) C r x) Dw r F w, x) = Aw, x) 15) for some power series C 0 x),..., C m x) Q[[x]] and Aw, x) Qw)[[x]] with A0, x) 0. Then m 1 ) C s x) Dw s MF w, x) = 1 Aw, x), 16) w s=0 where C s x) := m r r=s+1 s+1) Cr x) D r 1 s F 0, x). Proof: Apply the lemma with c r t) = C r e t ), ft) = F 0, e t ), gt) = e wt F w, e t ), at) = e wt Aw, e t ), noting that then ht) = we wt MF w, e t ). 5
6 2.2. Proof of Theorems 1 and 2 For the proof of 2) and 3), it is convenient to define F p w, x) also for p = 1. Set F 1 w, x) = w D 1 w Fw, x) = d=0 x d r=nd 1 r=d nw+r) P. 17) w+r) n wn) We have F 1 0, x)=1 and w 1 D w F 1 =F, so F p = M p+1 F 1 for all p 0, justifying the notation. It is straightforward to check that F 1 is a solution of the differential equation D n w x This has the form of 15) with F = F 1, A = w n F 1, m = n, and n 1 ndw + j )) F 1 = w n F 1. 18) j=0 C n x) = 1 n n x, C r x) = n r S n r n 1) x 0 < r < n), C 0 x) = 0, 19) where S n r n 1) as before denotes the n r)-th elementary symmetric function of 1, 2,..., n 1. Applying Corollary 1 repeatedly, we obtain n 1 p s=0 C p) s x) D s wf p w, x) = w n p 1 F 1 w, x) 0 p n 1), 20) where C s 0) x) = C s+1 x) with C r x) as in 19) and C s p) for p > 0 is given inductively by C p) s = n p r=s+1 ) r C r p 1) x) D r 1 s I p 1 x). 21) s + 1 In particular, by induction on p we find that the first two coefficients in 20) are given by p 1 C p) n 1 p = 1 nn x) I r x), 22) C p) n 2 p = nn n 1) p 1 x + 1 n n x) n r 1) I rx) 2 I r x) Equations 20) and 22) for p = n 1 give 1 n n x) n 2 ) p 1 I r x). 23) I r x) F n 1 w, x) = F 1 w, x). 24) Setting w = 0 in this relation and using F 1 0, x) = 1 gives equation 2). Then substituting 2) back into 24) gives F n 1 /I n 1 = F 1 and hence, applying w 1 D w to both sides, F n = F, proving Theorem 1. Similarly, taking p = n 2 in equations 20), 22), and 23) and then setting w = 0 gives n 2 n r 1) I rx) I r x) = n n n x 1 n n x,
7 and integrating this and exponentiating gives 3). Finally, we must prove the reflection symmetry 4). For this purpose, it is useful to construct the power series I p in another way. Define a function F 0 P by F 0 w, x) = d=0 x d r=nd nw+r) r=d w+r)n 25) and set F p w, x) = M p F0 w, x) for all p 0. Since F 0 w, x) is congruent to Fw, x) modulo w n, we find by induction on p that F p w, x) is congruent to F p w, x) modulo w n p for all 0 p n 1 and hence that I p x) = F p 0, x) in this range. We now argue as above, using F 0 instead of F 1. This function satisfies the differential equation n 1 Dw n 1 nx ndw + j )) F0 = w n 1. Applying Corollary 1 repeatedly, we obtain n 1 p s=0 for 0 p n 1, where the coefficients being given by j=1 C s p) x) Dw s F p w, x) = w n p 1 p) C s x) Q[[x]] can be calculated recursively, the top one C p) n 1 p x) = 1 nn x)i 0 x) I p 1 x). Specializing to p = n 1 and using 2), we find that F n 1 w, x) = I n 1 x) is independent of w. Now by downwards induction on p, using the equation F p = I p wdw 1 F p+1, we can reconstruct all of the power series F p w, x) n 1 p 0) from their special values I p x) = F p 0, x) at w = 0, obtaining in particular the formula w 1 n F0 w, x) = I 0 Dw 1 I 1 Dw 1 I n 2 Dw 1 I n 1 for the initial series F 0. Comparing the coefficients of x d on both sides of this equation, we find n 1 nd nw + r) [ww + 1) w + d)] n = d 0,...,d n 1 0 d 0 + +d n 1 =d c 0 d 0 ) c n 1 d n 1 ) w + d d n 1 )w + d d n 1 ) w + d n 1 ) for all d 0, where c p d) denotes the coefficient of x d in I p x). Splitting up the sum on the right into the subsum over n-tuples d 0,..., d n 1 ) with max{d r } d 1 and the sum over the n-tuples which are permutations of d, 0,..., 0), and using that c p 0) = 1 for all p, we can rewrite this equation as n 1 p=0 c p d) w n p 1 w + d) p = nd nw + r) n d w + r)n 0 d 0,...,d n 1 <d d 0 + +d n 1 =d c 0 d 0 ) c n 1 d n 1 ) w + d d n 1 ) w + d n 1 ) Now suppose by induction that c p d ) = c n p 1 d ) for all d < d and all 0 p n 1. Notice that this is true for d = 0 because c p 0) = I p 0) = 1 for all p, providing the starting point for the induction.) Then both terms on the right are 1) n 1 -invariant under the map w w d, as one sees for the second term by making the renumbering d r d n 1 r. It follows that the left-hand side has the same invariance and hence that c p d) = c n 1 p d) for all 0 p n 1, completing the inductive proof of the desired symmetry I n 1 p = I p. 7
8 2.3. Proof of Theorem 3 We now turn to the expansion of Fw, x) near w =. We first prove Lemma 1, which said that any periodic fixed point of the map M : P P has a logarithm which belongs to w Q[[x, w 1 ]]. Proof of Lemma 1: The effect of M on logarithms is given by M e Hw,x)) = e H w,x), where ) H DHw, x) DH0, x) w, x) = Hw, x) H0, x) + log 1 + ; 26) w here, as before, D denotes x x. Suppose that Hw, x) := log F w, x) is not Ow), and let e be the smallest integer such that the coefficient of x e in Hw, x) is not Ow) as w. Then Hw, x) = Cx e w N + xo w w) + x e O w w N 1 ) + Ox e+1 ) w ) 27) for some C 0 and N 2, where O w w ν ) denotes a polynomial in x with coefficients that grow at most like w ν as w and Ox e+1 ) denotes an element of x e+1 Qw)[[x]]. From 26) and 27), H w, x) = Hw, x) + Cex e w N 1 + xo w 1) + x e O w w N 2 ) + Ox e+1 ). This has the same form as 27) with the same C, e, and N. Iterating, we find that log M k F w, x) ) = Hw, x) + kcex e w N 1 + xo w 1) + x e O w w N 2 ) + Ox e+1 ), and this contradicts the assumption that M k F = F, since C 0 and N 2. As already mentioned in the introduction, Lemma 1 together with Theorem 1 implies that Fw, x) has an asymptotic expansion of the form 5). From the proof of the lemma, we see that each F p w, x) = M p Fw, x) has an asymptotic expansion F p w, x) e µx)w s=0 Φ p,s x) w s w ) 28) of the same form, with the same function µx) in the exponent. The equation F p+1 =MF p gives Φ 0,s = Φ s, Φ p+1,s = 1 + µ I p Φ p,s + { Φp,s 1 /I p ) if s 1, 0 otherwise, 29) where f denotes Df = x df/dx. We want to solve these equations by induction on p for small s. Before doing this, we begin with the following observation. Let Lx) = 1 n n x) 1/n as in Theorem 3. Then 2) says that the product of the functions I p x)/lx) p Z/nZ) equals 1, so if we define Lx) p H p x) = p 0), 30) I 0 x) I p 1 x) then we have the properties H 0 = 1, H p /H p+1 = I p /L, H 1 H 2 H n = 1, H p+n = H p, H n p = H 1 p, 31) where the last equality is originally true for 0 p n but then, in view of the periodicity of {H p }, holds for any p Z/nZ. A number of identities below are simpler to state in terms of the functions H p x) than in terms of the original functions I p x). 8
9 The case s = 0 of 29) gives by induction the formula Φ p,0 = 1 + µ ) p /I 0 I p 1. Combining this with the formulas F n = F and 2), we obtain 1 + µ ) n = L n, from which the first equation in 6) follows since µx) is a power series in x with no constant term. This also gives us the formula Φ p,0 x) = H p x) Φ 0 x) for all p 0, with H p as in 30). Now substituting this into the case s = 1 of 29) we find inductively Φ p,1 x) = H p x) Φ 1 x) + p Φ 0 L L + Φ 0 L p H r ) H r for all p 0. Setting p = n in this relation and using the third and fourth of equations 31) and F n = F, we deduce that Φ 0 = L, which is the second assertion of Theorem 3. At the same time we can refine the last two equations to p H Φ p,0 = H p L, Φ p,1 = H p Φ ) r 1 + p 0). 32) H r The proof of the third identity in 6) is similar, but the calculations are more complicated. The case s = 2 of 29) gives by induction the formula Φ p,2 = H p Φ 2 + p Φ1 ) p H ) r Φ1 + L H r L + 1 L for all p 0. Taking p = n, observing that n s 1 H r H s n 1 H r H s 2 s=2 H p H p=1 p ) 2 p s 1 s=2 n H ) ) 2 p = 1 2 p=1 by the third equation in 31), and using F n = F, we find that H p H r H s 1 p 1 + H r H s L n H p p=1 H p ) ) p r) H r ) 2 H r n Φ1 L ) = 1 2L n H ) 2 p + p=1 H p n 1 1 L p=0 ) p H p n + 1)n 2) = L n 1 ), H p 24 the last equation being Lemma 3 below. Integrating and using Φ 1 0) = 0 gives the last identity in 6). Lemma 3. The functions {H p x)} p Z/nZ satisfy 1 2L p mod n) H p H p ) 2 n + 1)n 2) = L n n 1 ) p H p. 33) 24 L H p=0 p The proof consists of expressing the left-hand side of 33) in terms of the functions I 0, I 1,..., I n 1 and their derivatives, getting rid of all square terms via the product rule, and then eliminating I n 1, I n 2, and I n 3. The last elimination is achieved by computing the coefficients C p n 3 p) inductively by 20), starting with C 0) n 3 = n 1)n 2)3n 1) nn 2 S 2 n 1) x = L /L n+1, 24 9
10 and then setting p = n 3, exactly as we did with C p n 1 p) and C p n 2 p) in Subsection 2.2 to prove eqs. 2) and 3). At this stage, all terms involving products of two functions I p cancel, and the resulting expression can be integrated. We omit the details, which are somewhat tedious, since the last identity in 6) also follows easily from Theorem Proof of Theorem 4 We set X = L n and Y = L n 1)/n. Note that Dµ) = L 1, DL) = LY, DX) = X 2 X, DY ) = XY. 34) The first identity implies that D w e µw = e µw Dw, where D w = D + Lw. By induction on k, the powers of the differential operator D w are given by D k w = k m=0 ) k D w m 1) D k m m = D k + k Lw D k 1 + A second induction gives the formula kk 1) 2 Lw) 2 + Y Lw) ) D k ) D m w 1) = m H m,j X) Lw) m j, 36) j=0 with H m,j Z[X, Y ] Q[X] given by 7). The function Fw, x) satisfies the ODE D n w w n x n ndw + j )) F = 0. j=1 Since D w e µw = e µw Dw, the function Fw, x) = e µx)w Fw, x) satisfies the differential equation L F = 0, where L is the differential operator L = L n D n w w n x n n Dw + j )) j=1 = D n w Lw) n L n 1) n S r n) n r Dn r w. Using 35) and 36), we can expand L as L = n k=1 Lw)n k L k, with L k defined by 9). Combining the differential equation L F = 0 with the asymptotic expansion Fw, x) s 0 Φ sx)w s for large w, we obtain 12). We will next use 12) to prove by induction that Φ s belongs to LQ[L]. Since L 1 LQ[L]) = L 2 Y Q[L], it suffices to show that L k LQ[L]) L k+1 Y Q[L] 2 k n). 37) 10
11 Let I Q[L] be the ideal generated by XY. Since D and Y commute modulo I by 34) and since D ry )L r = 0, we have { D Y )D 2Y )... D ky )L r L r I if 1 r k, L r Y Q[L] if r k + 1. Therefore 37) is a consequence of the following lemma. Lemma 4. For all k > 1, L k ) n D Y )D 2Y ) D ky ) mod I). k Proof: The recursion 7) for H m,j shows that H m,j h m,j Y j mod I), where h m,j Z is given recursively by h 0,j = δ 0,j, h m,j = h m 1,j + m j)h m 1,j 1 m 1 38) with h m 1,j 1 = 0 for j = 0). Thus h m,j = S m j) m, where S m k) denotes a Stirling number of the second kind the number of ways of partitioning a set of m elements into k non-empty subsets). We also note 1 L n )n r 1) r Y r mod I) for all r 1. Combining these facts with 9), we find that L k k i=0 k i ) ) n r 1) r S r n) S n k) n r i Y k i D i mod I). i The desired congruence for L k now follows from the generating series calculation k i=0 = k i ) n r 1) r i n n i ) n r 1) r i i=0 j=0 ) S r n) S n k) n r i t i [ S r n) 1 n k)! n k j=0 1) n k j n k j i=0 )j n r i ]) n k 1 ) n n r n k ) n r = 1) n k j 1) r S r n) j n r i t i n k)! j i n k 1 ) n n k = 1) n k j 1) r S r n)j + t) n r n k)! j j=0 n k n! ) ) n k j + t 1 = 1) n k j n k)! j n j=0 ) ) n! t 1 n = = t 1)t 2) t k), n k)! k k where the first equality follows from the well-known fact that the expression in square brackets equals S n k) n r i if i + r k and 0 for i + r > k and the second-to-last equality is obtained by expanding 1 + u) t u) 1) t 1 by the binomial theorem and equating coefficients of t n. This completes the proof of part ii) of Theorem 4. Part iii) of Theorem 4 follows from the differential equation 12) by induction on s. 11 t i
12 3. Further discussion of the large w expansion of Fw, x) In this final section we give some further information and conjectures about the power series Φ s x) defined by equation 5). We begin by giving the numerical values for n 5 and s 4. For this purpose it is convenient to divide Φ s /L by n 2)n + 1)/24n) s /s! and write the result as the sum of 1 L n 1 ) s and a correction term, because the formulas then become much simpler than without this renormalization: n = 3 : s = 1 : 1 L 2 s = 2 : 1 L 2 ) 2 s = 3 : 1 L 2 ) L 3 + 4L 6 ) s = 4 : 1 L 2 ) L 2 5L L 5 + 4L 6 151L 8 ) n = 4 : s = 1 : 1 L 3 s = 2 : 1 L 3 ) 2 s = 3 : 1 L 3 ) L 297L L 9 ) n = 4 1 L 3 ) L 20L L L L L L 12 ) n = 5 : s = 1 : 1 L 4 s = 2 1 L 4 ) 2 s = 3 : 1 L 4 ) L2 504L L 12 ) s = 4 : 1 L 4 ) L L2 168L L L L L L 16 ) Table: List of values of s! 24n sφs n 2)n+1)) /L for s = 1, 2, 3, 4 and n = 3, 4, 5 This suggests that the series s Φ s/l) w s is given to a first approximation by a pure exponential exp n 2)n+1) 24n 1 L n 1 )/w ) and hence that the formulas for the coefficients of the expansion 5) may become simpler if we take the logarithm. Doing this, we find an expansion which begins log Fw, x) = µx)w + log Lx) + n 2)n + 1)1 Lx)n 1 ) 24n w w 2 + and in which, at least experimentally, the coefficient of w j for j 1 is the sum of a term independent of x and a term of the form L j times a polynomial without constant term) in L n. By applying the operator w 1 D and adding 1, this can be stated more elegantly as follows. Conjecture: If F is given by 1), then 1 + x w x log Fw, x)? = L k=0 P k n, L n ) nlw) k, 39) where P k n, X) is a polynomial in X of degree k with coefficients in Q[n]. We have verified this conjecture up to order O w 6), with the values of the corresponding coef- 12
13 ficients P k being given by where P 0 n, X) = 1, P 1 n, X) = X 1, n + 1)n 1)n 2) P 2 n, X) = X 1)X, 24 P 3 n, X) = 0, n + 1)n 1)n 2) P 4 n, X) = X 1)A 3 X 3 + A 2 X 2 + A 1 X), 5760 n + 1)n 1)n 2) P 5 n, X) = X 1)B 4 X 4 + B 3 X 3 + B 2 X 2 + B 1 X), 5760 A 1 = n 3)7n 3 17n n 24), A 2 = 2n 3)3n 1)7n 2 9n + 14), A 3 = 3 14n 4 33n n 2 23n 2), B 1 = n 3)n 4)7n 3 17n n 24), B 2 = 2 n 1)n 2)49n 3 115n n 124), B 3 = 4 n 1)3n 1)3n 4)7n 2 9n + 14), B 4 = 8 n 1)3n 2)7n 3 11n n 1). The coefficients of the polynomials P k follow no apparent pattern apart from the divisibility by n + 1)n 1)n 2)XX 1): the common factors of A 1 and B 1 and of A 2 and B 3 are striking, but nothing similar occurs for the next two polynomials. On the other hand, there is a simple formula for the leading coefficient of P k n, X) with respect to n, namely at least up to k = 7) { α j e k X) n 4j 1 + On 4j 2 ) if k = 2j > 0, P k n, X) = j 1)α j e k X) n 4j + On 4j 1 ) if k = 2j + 1, where α j denotes the coefficient of u 2j u/2 in sinh u/2 α 0 = 1, α 1 = 1 24, α 2 = , α 3 = ,... ) and where e 1 = X 1, e 2 = X 2 X, e 3 = 2X 3 3X 2 + X,... are the polynomials defined by e k X) = k l=1 1) k l l 1)! S l) k Xl Z[X] with S l) k as before a Stirling number of the second kind. This is interesting because the argument X = L n of P k n, X) in equation 39) is in fact 1 n n x) 1 and the functions e k 1 x) 1 ) have the basic property ) 1 e k = d k 1 x d x Z[[x]] k 1). 1 x d=1 There is also a possible intriguing connection with modular and elliptic functions since, for example, the power series in two variables α j e 1 2j 1 x) u 2j 1 is closely related to the expansion of the Weierstrass -function and related Jacobi forms. This suggests possible hidden modularity properties of the original function Fw, x). 13
14 As a final remark, we observe that 39), if it is true, defines the power series Fw, x) even for non-integral values of n and shows that this function is analytic in n as well as in w and x. This seems surprising since F is defined as a hypergeometric function of order n and we would usually not expect such series to have a reasonable interpolation with respect to the order of the differential equation which they satisfy. References [BCOV] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic Anomalies in Topological Field Theories, Nucl. Phys. B ), [CaDGP] P. Candelas, X. de la Ossa, P. Green, L. Parkes, A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory, Nuclear Phys. B ), [Gi] A. Givental, The Mirror Formula for Quintic Threefolds, Amer. Math. Soc. Transl. Ser. 2, ). [Z1] A. Zinger, The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces, math/ [Z2] A. Zinger, Standard vs. Reduced Genus-One Gromov-Witten Invariants, math/ Max-Planck-Institut für Mathematik, Bonn zagier@mpim-bonn.mpg.de Department of Mathematics, SUNY Stony Brook, NY azinger@math.sunysb.edu 14
Some properties of hypergeometric series associated with mirror symmetry
Some properties of hypergeometric series associated with mirror symmetry Don Zagier and Aleksey Zinger June 28, 2011 Abstract We show that certain hypergeometric series used to formulate mirror symmetry
More informationarxiv: v1 [math.ag] 15 Jul 2018
. NOTE ON EQUIVARIANT I-FUNCTION OF LOCAL P n HYENHO LHO arxiv:1807.05501v1 [math.ag] 15 Jul 2018 Abstract. Several properties of a hyepergeometric series related to Gromov-Witten theory of some Calabi-Yau
More informationEnumerative Geometry: from Classical to Modern
: from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:
More informationOn the BCOV Conjecture
Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called
More informationAsymptotic of Enumerative Invariants in CP 2
Peking Mathematical Journal https://doi.org/.7/s4543-8-4-4 ORIGINAL ARTICLE Asymptotic of Enumerative Invariants in CP Gang Tian Dongyi Wei Received: 8 March 8 / Revised: 3 July 8 / Accepted: 5 July 8
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More information* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More information(n = 0, 1, 2,... ). (2)
Bull. Austral. Math. Soc. 84(2011), no. 1, 153 158. ON A CURIOUS PROPERTY OF BELL NUMBERS Zhi-Wei Sun and Don Zagier Abstract. In this paper we derive congruences expressing Bell numbers and derangement
More informationGroup, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,
Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ
More informationGeometry of the Calabi-Yau Moduli
Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
More informationChapter Generating Functions
Chapter 8.1.1-8.1.2. Generating Functions Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 8. Generating Functions Math 184A / Fall 2017 1 / 63 Ordinary Generating Functions (OGF) Let a n (n = 0, 1,...)
More informationPredictive criteria for the representation of primes by binary quadratic forms
ACTA ARITHMETICA LXX3 (1995) Predictive criteria for the representation of primes by binary quadratic forms by Joseph B Muskat (Ramat-Gan), Blair K Spearman (Kelowna, BC) and Kenneth S Williams (Ottawa,
More informationCounting curves on a surface
Counting curves on a surface Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo University of Pennsylvania, May 6, 2005 Enumerative geometry Specialization
More informationMoreover this binary operation satisfies the following properties
Contents 1 Algebraic structures 1 1.1 Group........................................... 1 1.1.1 Definitions and examples............................. 1 1.1.2 Subgroup.....................................
More informationMATH 361: NUMBER THEORY FOURTH LECTURE
MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationPeriods and generating functions in Algebraic Geometry
Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted
More informationA field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:
Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under
More informationElementary Algebra Chinese Remainder Theorem Euclidean Algorithm
Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. This is presented in more depth than what we
More informationSECOND-ORDER RECURRENCES. Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C
p-stability OF DEGENERATE SECOND-ORDER RECURRENCES Lawrence Somer Department of Mathematics, Catholic University of America, Washington, D.C. 20064 Walter Carlip Department of Mathematics and Computer
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More informationFinite Fields: An introduction through exercises Jonathan Buss Spring 2014
Finite Fields: An introduction through exercises Jonathan Buss Spring 2014 A typical course in abstract algebra starts with groups, and then moves on to rings, vector spaces, fields, etc. This sequence
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More informationPUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.
PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice
More informationThe density of rational points on non-singular hypersurfaces, I
The density of rational points on non-singular hypersurfaces, I T.D. Browning 1 and D.R. Heath-Brown 2 1 School of Mathematics, Bristol University, Bristol BS8 1TW 2 Mathematical Institute,24 29 St. Giles,Oxford
More informationCurve counting and generating functions
Curve counting and generating functions Ragni Piene Università di Roma Tor Vergata February 26, 2010 Count partitions Let n be an integer. How many ways can you write n as a sum of two positive integers?
More informationArithmetic Mirror Symmetry
Arithmetic Mirror Symmetry Daqing Wan April 15, 2005 Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China Department of Mathematics, University of California, Irvine, CA 92697-3875
More informationwhere c R and the content of f is one. 1
9. Gauss Lemma Obviously it would be nice to have some more general methods of proving that a given polynomial is irreducible. The first is rather beautiful and due to Gauss. The basic idea is as follows.
More informationOn the Frobenius Numbers of Symmetric Groups
Journal of Algebra 221, 551 561 1999 Article ID jabr.1999.7992, available online at http://www.idealibrary.com on On the Frobenius Numbers of Symmetric Groups Yugen Takegahara Muroran Institute of Technology,
More informationTHE MODULAR CURVE X O (169) AND RATIONAL ISOGENY
THE MODULAR CURVE X O (169) AND RATIONAL ISOGENY M. A. KENKU 1. Introduction Let N be an integer ^ 1. The affine modular curve Y 0 (N) parametrizes isomorphism classes of pairs (E ; C N ) where E is an
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationAhlswede Khachatrian Theorems: Weighted, Infinite, and Hamming
Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Yuval Filmus April 4, 2017 Abstract The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of
More informationCOUNTING NUMERICAL SEMIGROUPS BY GENUS AND SOME CASES OF A QUESTION OF WILF
COUNTING NUMERICAL SEMIGROUPS BY GENUS AND SOME CASES OF A QUESTION OF WILF NATHAN KAPLAN Abstract. The genus of a numerical semigroup is the size of its complement. In this paper we will prove some results
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)
More informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationRoots of Unity, Cyclotomic Polynomials and Applications
Swiss Mathematical Olympiad smo osm Roots of Unity, Cyclotomic Polynomials and Applications The task to be done here is to give an introduction to the topics in the title. This paper is neither complete
More informationA connection between number theory and linear algebra
A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationGENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui
GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli
More informationUpper Bounds for Partitions into k-th Powers Elementary Methods
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 9, 433-438 Upper Bounds for Partitions into -th Powers Elementary Methods Rafael Jaimczu División Matemática, Universidad Nacional de Luján Buenos Aires,
More informationFormal groups. Peter Bruin 2 March 2006
Formal groups Peter Bruin 2 March 2006 0. Introduction The topic of formal groups becomes important when we want to deal with reduction of elliptic curves. Let R be a discrete valuation ring with field
More informationComputing a Lower Bound for the Canonical Height on Elliptic Curves over Q
Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q John Cremona 1 and Samir Siksek 2 1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationN = 2 String Amplitudes *
LBL-37660 August 23, 1995 UCB-PTH-95/30 N = 2 String Amplitudes * Hirosi Oogurit Theoretical Physics Group Lawrence Berkeley Laboratory University of California Berkeley, California 94 720 To appear in
More informationThe cocycle lattice of binary matroids
Published in: Europ. J. Comb. 14 (1993), 241 250. The cocycle lattice of binary matroids László Lovász Eötvös University, Budapest, Hungary, H-1088 Princeton University, Princeton, NJ 08544 Ákos Seress*
More informationComputations/Applications
Computations/Applications 1. Find the inverse of x + 1 in the ring F 5 [x]/(x 3 1). Solution: We use the Euclidean Algorithm: x 3 1 (x + 1)(x + 4x + 1) + 3 (x + 1) 3(x + ) + 0. Thus 3 (x 3 1) + (x + 1)(4x
More informationARITHMETIC PROGRESSIONS IN CYCLES OF QUADRATIC POLYNOMIALS
ARITHMETIC PROGRESSIONS IN CYCLES OF QUADRATIC POLYNOMIALS TIMO ERKAMA It is an open question whether n-cycles of complex quadratic polynomials can be contained in the field Q(i) of complex rational numbers
More informationCourse MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography
Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More information#A42 INTEGERS 10 (2010), ON THE ITERATION OF A FUNCTION RELATED TO EULER S
#A42 INTEGERS 10 (2010), 497-515 ON THE ITERATION OF A FUNCTION RELATED TO EULER S φ-function Joshua Harrington Department of Mathematics, University of South Carolina, Columbia, SC 29208 jh3293@yahoo.com
More informationERIC LARSON AND LARRY ROLEN
PROGRESS TOWARDS COUNTING D 5 QUINTIC FIELDS ERIC LARSON AND LARRY ROLEN Abstract. Let N5, D 5, X) be the number of quintic number fields whose Galois closure has Galois group D 5 and whose discriminant
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18. PIDs Definition 1 A principal ideal domain (PID) is an integral
More informationLecture Notes on DISCRETE MATHEMATICS. Eusebius Doedel
Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel c Eusebius J. Doedel, 009 Contents Logic. Introduction............................................................................... Basic logical
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationConsidering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.
Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and
More informationSum Theorems for Multiple Zeta Values, Old and New. Michael E. Hoffman. Number Theory Talk Max-Planck-Institut für Mathematik, Bonn 29 June 2016
for Zeta, Michael E. Hoffman U. S. Naval Academy Number Theory Talk Max-Planck-Institut für Mathematik, Bonn 29 June 2016 1 2 3 4 5 6 For positive integers a 1,..., a k with a 1 > 1 we define the corresponding
More informationSOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM
Acta Arith. 183(018), no. 4, 339 36. SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM YU-CHEN SUN AND ZHI-WEI SUN Abstract. Lagrange s four squares theorem is a classical theorem in number theory. Recently,
More information1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.
1 2 3 style total Math 415 Examination 3 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. The rings
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationarxiv: v2 [math.nt] 23 Sep 2011
ELLIPTIC DIVISIBILITY SEQUENCES, SQUARES AND CUBES arxiv:1101.3839v2 [math.nt] 23 Sep 2011 Abstract. Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences
More informationHomework 10 M 373K by Mark Lindberg (mal4549)
Homework 10 M 373K by Mark Lindberg (mal4549) 1. Artin, Chapter 11, Exercise 1.1. Prove that 7 + 3 2 and 3 + 5 are algebraic numbers. To do this, we must provide a polynomial with integer coefficients
More informationPoints of Finite Order
Points of Finite Order Alex Tao 23 June 2008 1 Points of Order Two and Three If G is a group with respect to multiplication and g is an element of G then the order of g is the minimum positive integer
More informationChapter 1 The Real Numbers
Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus
More informationCDM. Finite Fields. Klaus Sutner Carnegie Mellon University. Fall 2018
CDM Finite Fields Klaus Sutner Carnegie Mellon University Fall 2018 1 Ideals The Structure theorem Where Are We? 3 We know that every finite field carries two apparently separate structures: additive and
More informationMath 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction
Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationCounting problems in Number Theory and Physics
Counting problems in Number Theory and Physics Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Encontro conjunto CBPF-IMPA, 2011 A documentary on string theory
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationA New Shuffle Convolution for Multiple Zeta Values
January 19, 2004 A New Shuffle Convolution for Multiple Zeta Values Ae Ja Yee 1 yee@math.psu.edu The Pennsylvania State University, Department of Mathematics, University Park, PA 16802 1 Introduction As
More informationSYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS
SYMMETRY AND SPECIALIZABILITY IN THE CONTINUED FRACTION EXPANSIONS OF SOME INFINITE PRODUCTS J MC LAUGHLIN Abstract Let fx Z[x] Set f 0x = x and for n 1 define f nx = ff n 1x We describe several infinite
More informationA Motivated Introduction to Modular Forms
May 3, 2006 Outline of talk: I. Motivating questions II. Ramanujan s τ function III. Theta Series IV. Congruent Number Problem V. My Research Old Questions... What can you say about the coefficients of
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationSOLUTIONS TO PROBLEM SET 1. Section = 2 3, 1. n n + 1. k(k + 1) k=1 k(k + 1) + 1 (n + 1)(n + 2) n + 2,
SOLUTIONS TO PROBLEM SET 1 Section 1.3 Exercise 4. We see that 1 1 2 = 1 2, 1 1 2 + 1 2 3 = 2 3, 1 1 2 + 1 2 3 + 1 3 4 = 3 4, and is reasonable to conjecture n k=1 We will prove this formula by induction.
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationNotes on Systems of Linear Congruences
MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the
More informationNotes on the Matrix-Tree theorem and Cayley s tree enumerator
Notes on the Matrix-Tree theorem and Cayley s tree enumerator 1 Cayley s tree enumerator Recall that the degree of a vertex in a tree (or in any graph) is the number of edges emanating from it We will
More informationCYCLOTOMIC POLYNOMIALS
CYCLOTOMIC POLYNOMIALS 1. The Derivative and Repeated Factors The usual definition of derivative in calculus involves the nonalgebraic notion of limit that requires a field such as R or C (or others) where
More informationCollatz cycles with few descents
ACTA ARITHMETICA XCII.2 (2000) Collatz cycles with few descents by T. Brox (Stuttgart) 1. Introduction. Let T : Z Z be the function defined by T (x) = x/2 if x is even, T (x) = (3x + 1)/2 if x is odd.
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,
More information11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic
11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic Bezout s Lemma Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we
More informationbe any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UFD. Therefore
More informationGroups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002
Background on Groups, Rings, and Finite Fields Andreas Klappenecker September 12, 2002 A thorough understanding of the Agrawal, Kayal, and Saxena primality test requires some tools from algebra and elementary
More informationCourse 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography
Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2006 Contents 9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups
More informationx i x j x l ωk x j dx i dx j,
Exterior Derivatives In this section we define a natural differential operator on smooth forms, called the exterior derivative. It is a generalization of the diffeential of a function. Motivations: Recall
More information2 More on Congruences
2 More on Congruences 2.1 Fermat s Theorem and Euler s Theorem definition 2.1 Let m be a positive integer. A set S = {x 0,x 1,,x m 1 x i Z} is called a complete residue system if x i x j (mod m) whenever
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationKnots and Mirror Symmetry. Mina Aganagic UC Berkeley
Knots and Mirror Symmetry Mina Aganagic UC Berkeley 1 Quantum physics has played a central role in answering the basic question in knot theory: When are two knots distinct? 2 Witten explained in 88, that
More informationHandout - Algebra Review
Algebraic Geometry Instructor: Mohamed Omar Handout - Algebra Review Sept 9 Math 176 Today will be a thorough review of the algebra prerequisites we will need throughout this course. Get through as much
More informationS ps S qs S rs S 0s. N pqr = s. S 2 2g
On generalizations of Verlinde's formula P. Bantay Inst. for Theor. Phys., Eotvos Univ. July, 000 Abstract It is shown that traces of mapping classes of finite order may be expressed by Verlinde-like formulae.
More information8. Dirichlet s Theorem and Farey Fractions
8 Dirichlet s Theorem and Farey Fractions We are concerned here with the approximation of real numbers by rational numbers, generalizations of this concept and various applications to problems in number
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationThe moduli space of binary quintics
The moduli space of binary quintics A.A.du Plessis and C.T.C.Wall November 10, 2005 1 Invariant theory From classical invariant theory (we refer to the version in [2]), we find that the ring of (SL 2 )invariants
More informationPolynomial evaluation and interpolation on special sets of points
Polynomial evaluation and interpolation on special sets of points Alin Bostan and Éric Schost Laboratoire STIX, École polytechnique, 91128 Palaiseau, France Abstract We give complexity estimates for the
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationThe primitive root theorem
The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under
More informationAlgorithmic Approach to Counting of Certain Types m-ary Partitions
Algorithmic Approach to Counting of Certain Types m-ary Partitions Valentin P. Bakoev Abstract Partitions of integers of the type m n as a sum of powers of m (the so called m-ary partitions) and their
More information1 Series Solutions Near Regular Singular Points
1 Series Solutions Near Regular Singular Points All of the work here will be directed toward finding series solutions of a second order linear homogeneous ordinary differential equation: P xy + Qxy + Rxy
More information