Euclidean Division for Multivariate Polynomials

Euclidean division for multivariate polynomials Anderson Beraldo de Araujo´ RA 065156 Disciplina Aneis´ e Corpos Prof. Dr. Fernando Eduardo Torres Orihuela Abstract In contrast to what is regularly done in the literature, this article extends Euclidean division to multivariate polynomials, preserving uniqueness of remainders. 1 Introduction In the context of polynomials over one variable, Euclidean division is the process of division of two polynomials, which produces a quotient and a remainder with a degree iqual to zero or smaller than the divisor. Its main property is that the quotient and remainder are unique polynomials, under some conditions. According to Brown (1973), the generalization of Euclidean division to multivariate polynomials has a long history; it is straigthforward, but with many pitfalls. Notably, in one-variable polynomials, Euclidean division is the base for Euclid’s algorithm, which permit us to compute greatest common divisors. Nonetheless, in the multivariate case this is not so. Euclidean domains holds Bezout’s identity: gcd(a;b) = ra+sb. This fails in multivariate polynomial rings F[x1;:::;xn], n ≥ 2, since gcd(x1;x2) = 1 but there is no Bezout equation 1 = x1 f + x2g (evaluating at x1 = 0 = x2 implies 1 = 0 in F). Moreover, the usual version of Euclidean division for multivariate polynomials does not guarantee the uniqueness of remainders. In this work, we will provide a simple variation of Euclidean division, as it is displayed in Cox et alli (2015, p. 61-68), preserving uniqueness of remainders. The fundamental idea is to use a quisksort algorithm to order the monomials before to make the division. This small change is sufficient to ensure the uniqueness of remainders, as it will be shown. In Section 2, we remind some concepts related to multivariate polynomials, doing an especial emphasis on the ordering over monomials. In Section 3, we exhibt the Euclidean divison over multivariate polynomials. To conclude, we make some remarks on future works. 1 2 Monomials ordering We are going to discuss polynomials in n variables x1;:::;xn with coefficients in an arbitrary field F. We start by defining monomials. Definition 2.1. A monomial in x1;:::;xn is a product of the form a1 a2 an x1 · x2 ···xn where all of the exponents a1;:::;an are nonnegative integers. The total degree of this monomial is the sum a1 + ··· + an. We can simplify the notation for monomials as follows: let a = (a1;:::;an) be an n-tuple of nonnegative integers. Then we set a a1 a2 an x = x1 · x2 ···xn a When a = (0;:::;0), note that x = 1. We also let jaj = a1 + ··· + an denote the total degree of the monomial xa . Definition 2.2. A polynomial f in x1;:::;xn with coefficients in a field F is a finite linear combination (with coefficients in F) of monomials. We will write a polynomial f in the form a f = ∑aa x a where the sum is over a finite number of n-tuples a = (a1;:::;an). The set of all polynomials in x1;:::;xn with coefficients in F is denoted F[x1;:::;xn]. When dealing with polynomials in a small number of variables, we will usually dispense with subscripts. Thus, polynomials in one, two, and three variables lie in F[x], F[x;y], and F[x;y;z], respectively. For example, 1 p = 2x3y2z + y3z3 − 3xyz + y2 2 is a polynomial in Q[x;y;z]. We will usually use the letters f , g, p, q, r to refer to polynomials. We will use the following terminology in dealing with polynomials. The definition of sums and products of multivariate polynomials is analogous to the one variable case. The sum and product of two polynomials is again a polynomial. We say that a polynomial f divides a polynomial g provided that f = gp for some polynomial p 2 F[x1;:::;xn] One can show that, under addition and multiplication, F[x1;:::;xn] satisfies all of the field axioms except for the existence of multiplicative inverses. This means that F[x1;:::;xn] is a polynomial ring. a Definition 2.3. Let p = ∑a aa x be a polynomial in F[x1;:::;xn]. We call aa the coefficient of the monomial xa . If a a , 0, then we call a xa a term of p. The degree of p , 0, denoted deg(p), is the maximum jaj such that the coefficient aa is nonzero. The total degree of the zero polynomial is undefined. 2 3 2 1 3 3 2 As an example, the polynomial p = 2x y z + 2 y z − 3xyz + y given above has four terms and degree six. Note that there are two terms of maximal degree, which is something that cannot happen for polynomials of one variable. For this reason, we need to order the terms of multivariate polynomials. For the division algorithm on polynomials in one variable, we are dealing with the degree ordering on the one-variable monomials: m+1 m ··· > x > x > ··· > x2 > x > 1: The success of the algorithm depends on working systematically with the leading terms in f and g, and not removing terms at random from f using arbitrary terms from g. First, we note that we can reconstruct the monomial xa = xa1 ···xan from the n-tuple of n exponents a = (a1;:::;an) 2 N in the sense that any ordering > we establish on the space Nn give us an ordering on monomials: if a > b according to this ordering, we will also say that xa > xb . Here the natural numbers N include the number 0. We also want our orderings to be compatible with the algebraic structure of polynomial rings. To begin, since a polynomial is a sum of monomials, we would like to be able to arrange the terms in a polynomial unambiguously in descending (or ascending) order. To do this, we must be able to compare every pair of monomials to establish their proper relative positions. Thus, we will require that our orderings be linear or total orderings. This means that for every pair of monomials xa and xb either xa > xb or xa = xb or xa < xb . A total order is also required to be transitive, so that xa > xb and xb > xg . Next, we must take into account the effect of the sum and product operations on polynomials. When we add polynomials, after combining like terms, we may simply rearrange the terms present into the appropriate order, so sums present no difficulties. Products are more subtle, however. Since multiplication in a polynomial ring distributes over addition, it suffices to consider what happens when we multiply a monomial times a polynomial. If doing this changed the relative ordering of terms, significant problems could result in any process similar to the Euclidean division in F[x], in which we must identify the leading terms in polynomials. The reason is that the leading term in the product could be different from the product of the monomial and the leading term of the original polynomial. Hence, we will require that all monomial orderings have the following additional property. If xa > xb and xg is any monomial, then we require that xa xg > xb xg . In terms of the exponent vectors, this property means that if xa > xb in our ordering on Nn , then, for all g 2 Nn, a +g > b +g. Finally, we will need that > is a well-ordering. This means that every nonempty subset of Nn has a smallest element under >. In other words, if A ⊆ Nn is nonempty, then there is a 2 A such that b > a or every b , a in A. With these considerations in mind, we make the following definition. n Definition 2.4. A monomial ordering on K[x1;:::;xn] is a relation > on N such that, first, > is a total ordering on Nn, second, if xa > xb and xg 2 Nn then xa xg > xb xg and, third, > is a well-ordering on Nn. Given a monomial ordering >, we say that a ≥ b when either a > b or a = b. Lemma 2.1. An order relation > on Nn is a well-ordering if and only if every strictly decreasing sequence in Nn a(1) > a(2) > a(3) > ··· eventually terminates. Proof. We will prove this in contrapositive form: > is not a well-ordering if and only if there is an infinite strictly decreasing sequence in Nn. If > is not a well-ordering, then some nonempty 3 subset S ⊆ Nn has no least element. Now pick a(1) 2 S. Since a(1) is not the least element, we can find a(1) > a(2) in S. Then a(2) is also not the least element, so that there is a(2) > a(3) in S. Continuing this way, we get an infinite strictly decreasing sequence a(1) > a(2) > a(3) > ···. Conversely, given such an infinite sequence, then fa(1);a(2);a(3);···g is a non-empty subset of Nn with no least element, and thus, > is not a well-ordering. In this work we will use an ordering on n-tuples called lexicographic order >lex. n Definition 2.5. Let a = (a1;:::;an) and b = (b1;:::;bn) be in N . We say a >lex b if the n a b leftmost nonzero entry of the vector difference a −b in N is positive. We will write x >lex x if a >lex b. Example 2.1. We have (1;2;0) >lex (0;3;4) since a − b = (1;−1;−4). On the other hand, (3;2;4) >lex (3;2;1) since a − b = (0;0;3).

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