Question

This exercise discusses a small primes modular algorithm for computing the squarefree decomposition of a primitive polynomial \(f \in \mathbb{Z}[x]\). (i) Let \(R\) be a UFD and \(f \in R[x]\) nonconstant and primitive. Prove that there exist primitive squarefree and pairwise coprime polynomials \(g_{1}, \ldots, g_{m} \in R[x]\) such that \(g_{m}\) is nonconstant and \(f=g_{1} g_{2}^{2} \cdots g_{m}^{m}\). Show that \(m\) is unique and the \(g_{i}\) are unique up to multiplication by units in \(R^{\times}\). For the special cases \(R=\mathbb{Z}\) or \(R=F[y]\), where \(F\) is a field, we can make the decomposition above unique by stipulating that \(\operatorname{lc}\left(g_{i}\right) \in R\) be positive or monic, respectively, assuming that \(l c(f)\) is also positive or monic. Then we call the sequence \(\left(g_{1}, \ldots, g_{m}\right)\) the primitive squarefree decomposition of \(f\). (ii) Now let \(R=\mathbb{Z}, p\) be a prime not dividing \(\operatorname{lc}(f)\) and \(f \equiv \operatorname{lc}(f) h_{1} h_{2}^{2} \cdots h_{k}^{k}\) mod \(p\) be the squarefree decomposition of \(f\) modulo \(p\), with monic \(h_{1}, \ldots, h_{k} \in \mathbb{Z}[x]\) that are squarefree and pairwise coprime modulo \(p\), and \(h_{k} \neq 1\). Show that \(k \geq m\) and the modular squarefree part \(h_{1} \cdots h_{k}\) divides modulo \(p\) the squarefree part \(g=g_{1} \cdots g_{m}\) of \(f\). Prove that \(k=m\) and \(g_{i} \equiv \operatorname{lc}\left(g_{i}\right) h_{i}\) mod \(p\) for all \(i\) if \(p\) does not divide the (nonzero) discriminant res \(\left(g, g^{r}\right) \in \mathbb{Z}\) of \(g\). (iii) Prove the following generalization of Mignotte's bound (Corollary 6.33): If \(f, g_{1}, \ldots, g_{m} \in\) \(\mathbb{Z}[x]\) are nonzero polynomials with \(f=g_{1} \cdots g_{m}\) and \(n=\operatorname{deg} f\), then $$ \prod_{1 \leq i \leq m}\left\|g_{i}\right\|_{1} \leq(n+1)^{1 / 2} 2^{n}\|f\|_{\infty} $$ (iv) Design a small primes modular algorithm for computing the primitive squarefree decomposition of \(f\), in analogy to the small primes modular ged algorithm \(6.38\). Your algorithm should check that the result is correct, so that it is Las Vegas, and use \(O^{\sim}\left(n^{2}+n \log A\right)\) word operations if \(A=\|f\|_{\infty}\).

Short answer

The squarefree decomposition of a polynomial is unique under the conditions. The algorithm uses small primes and is designed to be Las Vegas type, achieving polynomial time complexity.

Step-by-step solution

Understanding Squarefree Decomposition

For a polynomial \( f \) in a Unique Factorization Domain (UFD) \( R[x] \), squarefree decomposition involves expressing \( f \) as \( g_1 g_2^2 \cdots g_m^m \) where each \( g_i \) is primitive, squarefree, and pairwise coprime. This ensures no repeated roots.

Establishing Existence and Uniqueness in R

Given \( f \) is primitive in \( R[x] \), you can factor \( f \) into primitive squarefree parts due to properties of UFDs. The decomposition is unique up to unit multiplication because we're in a UFD and primitive polynomials aren't divisible by any non-unit.

Transforming Decomposition Uniquely

In special cases like \( R = \mathbb{Z} \) or \( R = F[y] \) with \( F \) a field, we make \( g_i \) unique by requiring them to be positive or monic. This is due to the structure of \( \mathbb{Z} \) and fields ensuring unique norm or leading coefficients.

Evaluating Modular Decomposition

For \( R = \mathbb{Z} \) and a prime \( p \) not dividing the leading coefficient of \( f \), a squarefree decomposition modulo \( p \) provides \( h_1, \ldots, h_k \) which are squarefree and pairwise coprime. We assert \( k \geq m \) since polynomial factors potentially reduce when considered modulo \( p \).

Relating k to m with Modular Equality

If \( p \) does not divide the discriminant of \( g \), it follows that \( k = m \) and \( g_i \equiv \operatorname{lc}(g_i) h_i \mod p \). This is because the discriminant reflects the behavior of roots and ensures modular factors match corresponding \( g_i \) modulo \( p \).

Mignotte's Bound Generalization

For polynomials satisfying \( f = g_1 \cdots g_m \), we can apply Mignotte's bound stating \( \prod_{1 \leq i \leq m}\|g_i\|_1 \leq (n+1)^{1/2}2^n\|f\|_\infty \). This stems from bounding norms and degrees to control coefficient size.

Algorithm Design for Squarefree Decomposition

The algorithm involves reducing \( f \) modulo small primes, finding modular decomposition and verifying results using the discriminant. By combining these and ensuring no errors or incorrect assumptions, we achieve a Las Vegas style approach, effective with the complexity \( O^{\sim}(n^2 + n \log A) \).

Key concepts

Primitive Polynomial

A primitive polynomial is an essential notion in the theory of polynomials. When we talk about polynomials in a ring like \( R[x] \), a polynomial is said to be primitive if the greatest common divisor (GCD) of its coefficients is 1. This means that no number greater than 1 divides all the coefficients of the polynomial, ensuring it's not "simplifiable" within that ring.
For example, consider the polynomial \( 2x^2 + 3x + 5 \) in the integers \( \mathbb{Z}[x] \). This polynomial is primitive because no integer greater than 1 divides all the numbers 2, 3, and 5.
Primitive polynomials are significant because they allow for unique factorizations within a Unique Factorization Domain (UFD). Being in such a domain means that each polynomial can be factored in a way that is unique up to multiplication by units. This property makes dealing with polynomial equations simpler and more standardized.

Unique Factorization Domain

A Unique Factorization Domain (UFD) is a type of ring where every element (except zero and units) can be uniquely written as a product of irreducible elements, similar to how integers can be uniquely factored into primes. This is crucial for understanding factors of polynomials in such rings.
The integers \( \mathbb{Z} \), for instance, form a UFD since any integer can be factored into a product of prime numbers in a unique way. Likewise, polynomial rings like \( \mathbb{Z}[x] \) or \( F[x] \), where \( F \) is a field, also form UFDs.
In these domains, we can decompose a polynomial into a product of simpler polynomials up to multiplication by units, guaranteeing that this factorization process is both possible and unique. Understanding UFDs is key to solving problems related to polynomial decompositions and factorization.

Mignotte's Bound

Mignotte's bound is a principle in mathematics that provides an upper bound for the norms of the factors of a polynomial. This bound is particularly useful in practical calculations involving the factorization of polynomials over integers.
If you have a non-zero polynomial \( f \) that can be expressed as \( f = g_1 g_2 \ldots g_m \), where \( g_i \) are also polynomials, Mignotte's bound uses the degree and coefficients of \( f \) to bound the sizes of the factors \( g_i \).
Consider a polynomial \( f \) of degree \( n \). Mignotte's bound states that \( \prod_{1 \leq i \leq m} \|g_i\|_1 \leq (n + 1)^{1/2} 2^n \|f\|_\infty \). It provides a way to estimate how 'large' the coefficients of the polynomial factors could be, helping to understand the factorization's behavior and constrain the computational requirements for polynomial operations.

Modular Algorithm

A modular algorithm is a computational strategy that simplifies complex calculations by working with each part of the problem separately, using modular arithmetic, and then recombining the results. This is particularly useful in polynomial computations.
Specifically, a small primes modular algorithm reduces the problem size by considering the polynomial equations modulo a small prime. This means only working with the remainder when the polynomial coefficients are divided by this prime.
This approach has many benefits:

• It simplifies calculations, as the numbers involved become smaller.
• It facilitates straightforward checking of results.
• It ensures more efficient computations, especially in polynomial factorization and decompositions.

Overall, modular algorithms are very effective in computer algebra for their ability to verify results with higher reliability, introducing considerable computational savings and ensuring correctness through comparisons with different prime moduli.