How Long to Read On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems)

How Long Does it Take to Read On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems)?

It takes the average reader 1 hour and 49 minutes to read On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems) by Joshua Erin Hill

Assuming a reading speed of 250 words per minute. Learn more

Description

We prove a combinatorial identity that relates the size of the value set of a map with the sizes of various iterated fiber products by this map. This identity is then used as the basis for several algorithms that calculate the size of the value set of a polynomial for a broad class of algebraic spaces, most generally an algorithm to calculate the size of the value set of a suitably well-behaved morphism between "nice" affine varieties defined over a finite field. In particular, these algorithms specialize to the case of calculating the size of the value set of a polynomial, viewed as a map between finite fields. These algorithms operate in deterministic polynomial time for fixed input polynomials (thus a fixed number of variables and polynomial degree), so long as the characteristic of the field grows suitably slowly as compared to the other parameters. Each of these algorithms also produces a fiber signature for the map, which for each positive integer j, specifies how many points in the image have fibers of cardinality exactly j. We adapt and analyze the zeta function calculation algorithms due to Lauder-Wan and Harvey, both as point counting algorithms and as algorithms for computation of one or many zeta functions. These value set cardinality calculation algorithms extend to amortized cost algorithms that offer dramatic computational complexity advantages, when the computational cost is amortized over all the results produced. The last of these amortized algorithms partially answers a conjecture of Wan, as it operates in time that is polynomial in log q per value set cardinality calculated. For the value set counting algorithms, these are the first such results, and offer a dramatic improvement over any previously known approach.

How long is On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems)?

On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems) by Joshua Erin Hill is 107 pages long, and a total of 27,499 words.

This makes it 36% the length of the average book. It also has 34% more words than the average book.

How Long Does it Take to Read On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems) Aloud?

The average oral reading speed is 183 words per minute. This means it takes 2 hours and 30 minutes to read On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems) aloud.

What Reading Level is On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems)?

On Calculating the Cardinality of the Value Set of a Polynomial (and Some Related Problems) is suitable for students ages 10 and up.

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