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# totalBetti -- a hash table containing the graded Betti numbers of a Veronese embedding

- Usage:
    `totalBetti(d,n,b)`
- Inputs:
  - d, an integer,
  - n, an integer,
  - b, an integer,
- Outputs:
  - a hash table,

## Description

This is a hash table for the total numbers of $\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. The keys of the hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the rank of $K_{p,q}(\mathbb{P}^n, d;b)$. This equals the Betti number $\beta_{p,p+q}(d,\mathbb{P}^n,b)$.Some tables are incomplete and we mark unknown entries with infinity.

Note that totalBetti differs from totalBettiTally only in that the output is a hash table instead of a Betti tally. One can convert the output of totalBetti into a Betti tally via the makeBettiTally function.

In example below we generate a hash table showing the total graded Betti numbers of $\mathbb{P}^{2}$ embedded by $\mathcal{O}(3)$.

    i1 : B = totalBetti(3,2,0)

    o1 = HashTable{(0, 0) => 1  }
                   (0, 1) => 0
                   (0, 2) => 0
                   (1, 0) => 0
                   (1, 1) => 27
                   (1, 2) => 0
                   (2, 0) => 0
                   (2, 1) => 105
                   (2, 2) => 0
                   (3, 0) => 0
                   (3, 1) => 189
                   (3, 2) => 0
                   (4, 0) => 0
                   (4, 1) => 189
                   (4, 2) => 0
                   (5, 0) => 0
                   (5, 1) => 105
                   (5, 2) => 0
                   (6, 0) => 0
                   (6, 1) => 27
                   (6, 2) => 0
                   (7, 0) => 0
                   (7, 1) => 0
                   (7, 2) => 1

    o1 : HashTable

If we wish to view these graded Betti numbers in the usual fashion, we can use makeBettiTally to convert the hash table above to a Betti tally.

    i2 : makeBettiTally B

                0  1   2   3   4   5  6 7
    o2 = total: 1 27 105 189 189 105 27 1
             0: 1  .   .   .   .   .  . .
             1: . 27 105 189 189 105 27 .
             2: .  .   .   .   .   .  . 1

    o2 : BettiTally

## Ways to use totalBetti:

- `totalBetti(ZZ,ZZ,ZZ)`

## For the programmer

The object totalBetti is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SchurVeronese/doc.m2:110:0.