Question

Let \(F\) be a field and \(n \in N\). What is the size of the sparse representation of the polynomial \(\Pi_{0 \leq i

Short answer

Sparse size is \(n+1\), and arithmetic circuit size is \(3n-2\).

Step-by-step solution

Understand the Polynomial

The polynomial given is \( \prod_{0 \leq i<n} (x + y^2) \). This is a product of \( n \) terms, each of which is \( (x + y^2) \). Expanding this product normally would result in \( (x + y^2)^n \).

Sparse Representation Size

To consider the sparse representation size, we need to count the total number of monomials with nonzero coefficients after this polynomial is expanded. However, \((x + y^2)^n\) implies a simple pattern in its expansion, since each term is a power of \(y^2\), not \(y\). Hence, there are \(n+1\) distinct monomials (i.e., each power of \(y^2\) from 0 to \(n\) combined with varying powers of \(x\)).

Check Polynomial Properties

Note that in \((x + y^2)^n\), we have powers of \(x\) ranging from \(n\) to 0, and powers of \( (y^2) \) ranging from \(0\) to \(n\). There will be only \(n+1\) terms, where each term is of the form \( \binom{n}{k} x^{n-k} (y^2)^k \).

Arithmetic Circuit Representation

To create an arithmetic circuit of size \(3n-2\), recognize that constructing each term involves a basic operation. \(y^2\) is computed once, then each \( (x + y^2) \) involves multiplying \(x\) and adding \(y^2\). For \(n\) terms: \(2n\) operations for additions and multiplications, plus \(n-1\) multiplications to combine the terms, gives \(2n + (n-1) = 3n-1\) which is efficient but needs additional optimal arrangement to fulfill the constraint. The circuit can be refined slightly but will represent each of these operations succinctly through optimal gate arrangement.

Key concepts

Arithmetic Circuit

An arithmetic circuit is a mathematical structure used to evaluate polynomials efficiently. It is a connected acyclic graph where nodes represent operations, such as addition or multiplication, and edges represent the flow of data. In the context of sparse polynomial representation, arithmetic circuits help in minimizing the number of operations needed.

For a polynomial like \((x + y^2)^n\), constructing an arithmetic circuit of size \(3n - 2\) involves strategically organizing the operations. Start by computing \(y^2\), which is simple to do once since it doesn't change. Each term in the polynomial is a product of \(x\) and an accumulating power of \(y^2\).

• Compute \(y^2\) initially (1 multiplication).
• For each multiplication or addition in \((x + y^2)^n\): manage \(2n\) operations directly from \(n\) terms.
• Combine the resulting terms with \(n-1\) multiplications.

Using this setup allows the representation size to be minimized, meeting efficiency requirements, while employing the fewest possible operations.

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial—or other polynomials—in a fully expanded form. When dealing with \((x + y^2)^n\), we look at each term in the expression and methodically expand it into the sum of terms with distinct monomials.

The polynomial \((x + y^2)^n\) specifically expands as a sum of terms: \( \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (y^2)^k \). This formula arises from the binomial theorem, which provides a systematic way to expand powers of binomials by looking at the combinations of terms. Each "\(x\)" and "\(y^2\)" interacts through combinations determined by coefficients \(\binom{n}{k}\).

• Terms in expanded forms are determined by powers of each base variable.
• Each monomial is derived by multiplying coefficients \(\binom{n}{k}\) with their respective powers.

This expansion leverages the symmetric properties of binomials and results in a manageable number of monomials, making it attractive for computations.

Monomial Counting

Monomial counting in the context of a polynomial like \((x + y^2)^n\) involves identifying how many unique terms exist after expansion. Each unique term is a monomial characterized by a combination of variables raised to specific powers.

For the given polynomial, observe that each monomial in \((x + y^2)^n\) corresponds to a distinct combination of \(x\) and \((y^2)\). Since \((x + y^2)\) expands into terms like \(\binom{n}{k} x^{n-k} (y^2)^k\), we only count distinct powers of \(x\) and \(y^2\).

• There are \(n+1\) distinct monomials because we have powers of \(y^2\) from 0 to \(n\).
• Each distinct monomial represents a combination of \(x^{n-k}\) and \((y^2)^k\).

This counting simplifies the evaluation, gives us a quick way to understand the polynomial's complexity, and helps in optimizing arithmetic circuits to handle the polynomial efficiently.