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# multiBetti -- a hash table containing the multigraded Betti numbers of a Veronese embedding

## Description

This function returns a hash table $H$ containing the multigraded Betti numbers for $\mathcal{O}(b)$ on $\mathbb{P}^{n}$ under the embedding by $d$-fold Veronese embedding given by $\mathcal{O}(d)$. The keys of the returned hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the the multigraded Betti decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. We record the multigraded Betti numbers via a multigraded Hilbert series. See Section 1.1 of [BEGY] for a more precise description of the multigraded Betti numbers of a Veronese embedding.

Note that the output of this function is sometimes enormous and so can take a long time to print on the screen.

 i1 : totalBettiTally(3,2,0) 0 1 2 3 4 5 6 7 o1 = total: 1 27 105 189 189 105 27 1 0: 1 . . . . . . . 1: . 27 105 189 189 105 27 . 2: . . . . . . . 1 o1 : BettiTally i2 : multiBetti(3,2,0) o2 = HashTable{(0, 0) => 1 } (0, 1) => 0 (0, 2) => 0 (1, 0) => 0 4 2 4 4 2 3 3 3 2 3 2 3 3 2 4 2 3 2 2 2 2 3 2 4 4 3 2 2 3 4 4 2 3 3 2 4 (1, 1) => T T + T T T + T T + T T + 2T T T + 2T T T + T T + T T + 2T T T + 3T T T + 2T T T + T T + T T T + 2T T T + 2T T T + T T T + T T + T T + T T 0 1 0 1 2 0 2 0 1 0 1 2 0 1 2 0 2 0 1 0 1 2 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 1 2 1 2 (1, 2) => 0 (2, 0) => 0 6 2 6 2 5 4 5 3 5 2 2 5 3 5 4 4 5 4 4 4 3 2 4 2 3 4 4 4 5 3 5 3 4 2 3 3 3 3 2 4 3 5 2 6 2 5 2 2 4 3 2 3 4 2 2 5 2 6 6 2 5 3 4 4 3 5 2 6 5 4 4 5 (2, 1) => T T T + T T T + T T + 3T T T + 4T T T + 3T T T + T T + T T + 4T T T + 7T T T + 7T T T + 4T T T + T T + 3T T T + 7T T T + 9T T T + 7T T T + 3T T T + T T T + 4T T T + 7T T T + 7T T T + 4T T T + T T T + T T T + 3T T T + 4T T T + 3T T T + T T T + T T + T T 0 1 2 0 1 2 0 1 0 1 2 0 1 2 0 1 2 0 2 0 1 0 1 2 0 1 2 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 1 2 (2, 2) => 0 (3, 0) => 0 7 4 7 3 2 7 2 3 7 4 6 5 6 4 2 6 3 3 6 2 4 6 5 5 6 5 5 2 5 4 3 5 3 4 5 2 5 5 6 4 7 4 6 2 4 5 3 4 4 4 4 3 5 4 2 6 4 7 3 7 2 3 6 3 3 5 4 3 4 5 3 3 6 3 2 7 2 7 3 2 6 4 2 5 5 2 4 6 2 3 7 7 4 6 5 5 6 4 7 (3, 1) => T T T + 2T T T + 2T T T + T T T + 2T T T + 5T T T + 7T T T + 5T T T + 2T T T + 2T T T + 7T T T + 12T T T + 12T T T + 7T T T + 2T T T + T T T + 5T T T + 12T T T + 15T T T + 12T T T + 5T T T + T T T + 2T T T + 7T T T + 12T T T + 12T T T + 7T T T + 2T T T + 2T T T + 5T T T + 7T T T + 5T T T + 2T T T + T T T + 2T T T + 2T T T + T T T 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 (3, 2) => 0 (4, 0) => 0 8 5 2 8 4 3 8 3 4 8 2 5 7 6 2 7 5 3 7 4 4 7 3 5 7 2 6 6 7 2 6 6 3 6 5 4 6 4 5 6 3 6 6 2 7 5 8 2 5 7 3 5 6 4 5 5 5 5 4 6 5 3 7 5 2 8 4 8 3 4 7 4 4 6 5 4 5 6 4 4 7 4 3 8 3 8 4 3 7 5 3 6 6 3 5 7 3 4 8 2 8 5 2 7 6 2 6 7 2 5 8 (4, 1) => T T T + 2T T T + 2T T T + T T T + 2T T T + 5T T T + 7T T T + 5T T T + 2T T T + 2T T T + 7T T T + 12T T T + 12T T T + 7T T T + 2T T T + T T T + 5T T T + 12T T T + 15T T T + 12T T T + 5T T T + T T T + 2T T T + 7T T T + 12T T T + 12T T T + 7T T T + 2T T T + 2T T T + 5T T T + 7T T T + 5T T T + 2T T T + T T T + 2T T T + 2T T T + T T T 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 (4, 2) => 0 (5, 0) => 0 9 5 4 9 4 5 8 7 3 8 6 4 8 5 5 8 4 6 8 3 7 7 8 3 7 7 4 7 6 5 7 5 6 7 4 7 7 3 8 6 8 4 6 7 5 6 6 6 6 5 7 6 4 8 5 9 4 5 8 5 5 7 6 5 6 7 5 5 8 5 4 9 4 9 5 4 8 6 4 7 7 4 6 8 4 5 9 3 8 7 3 7 8 (5, 1) => T T T + T T T + T T T + 3T T T + 4T T T + 3T T T + T T T + T T T + 4T T T + 7T T T + 7T T T + 4T T T + T T T + 3T T T + 7T T T + 9T T T + 7T T T + 3T T T + T T T + 4T T T + 7T T T + 7T T T + 4T T T + T T T + T T T + 3T T T + 4T T T + 3T T T + T T T + T T T + T T T 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 (5, 2) => 0 (6, 0) => 0 9 7 5 9 6 6 9 5 7 8 8 5 8 7 6 8 6 7 8 5 8 7 9 5 7 8 6 7 7 7 7 6 8 7 5 9 6 9 6 6 8 7 6 7 8 6 6 9 5 9 7 5 8 8 5 7 9 (6, 1) => T T T + T T T + T T T + T T T + 2T T T + 2T T T + T T T + T T T + 2T T T + 3T T T + 2T T T + T T T + T T T + 2T T T + 2T T T + T T T + T T T + T T T + T T T 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 (6, 2) => 0 (7, 0) => 0 (7, 1) => 0 9 9 9 (7, 2) => T T T 0 1 2 o2 : HashTable

## Ways to use multiBetti :

• multiBetti(ZZ,ZZ,ZZ)

## For the programmer

The object multiBetti is .