AMNote on Resultants

David Weinberg and Clyde Martin

Department of Mathematics TQX~S Tech University Lubbock, TQUIS 79409

Transmitted by John Casti

ABSTRACT

An explicit technique is developed for the calculation of the number of common zeros of a set of polynomials. The number of common zeros is determined by the vanishing of certain resultantlike polynomials.

1. INTRODUCTION

The following problem is classical: Find necessary and sufficient condi- tions, in terms of the coefficients, for a given finite collection of polynomials in one variable to have a common root. In the case of two polynomials these conditions are very explicit and are in the classical literature; see for example the presentation in Van der Waerden [5]. It is surprising that direct generali- zations of this result to the case of three or more polynomials have appeared only very recently. S. Bamett [l] generalizes the results of Vardulakis and Stoyle [6] to the case where not all polynomials have the same degree. In both [l] and [6] the results are presented by exhibiting a single matrix, of large dimension, formed from the coefficients of the polynomials, whose rank determines the number of common roots. In particular, if there are h + 1

polynomials with maximum degree m and next highest degree p, then [l] exhibits an (hm + p) x (m + p) matrix whose rank defect is equal to the number of common roots. Furthermore, it is proved in [l] that if this matrix is put into row echelon form using row transformations only, then the last nonvanishing row gives the coefficients of a g.c.d.

This paper presents a different approach to this same problem. This is useful because the problem arises in many different contexts and it is often necessary to have far more explicit techniques than the rank conditions

APPLIED MATHEMATKS AND COMPLJTATZON 24:303-309 (1987) 303

0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 009&3003/87/$03.50

304 DAVID WEINBERG AND CLYDE MARTIN

presented in [l] and [6]. The purpose of this note is to exhibit explicitly invariants, which are various sums of determinants, formed from the coeffi- cients, whose vanishing is necessary and sufficient for the collection of polynomials to have any given number of common roots. An application is then given to the complex invariant theory of a single polynomial.

In the case of h + 1 polynomials with maximum degree m, next highest degree p, and the minimum degree 12, the number of invariants required in this note for d common roots is

(d+1)( yy). The number of invariants for the same purpose implied by the rank defect condition in [l] is

d

Ii

hm+p m+p

it 1 j=. m+p-j j ’

which is much larger than the number calculated in this note.

2. CLASSICAL RESULTS

Our results are based on two classical theorems. The first is the well-known theorem on the resultant of two polynomials. We present it as stated in Dickson [3]. Let

f(x) = aoxm + UIP1 + . . * + a,,

g(x) =bor”+blx”-l+ ..* +b”.

Let

a0

0

0

a1

a0

0

R= iI . . .

b0 b, 0 b0

. . .

a2

a1

a0

0’

b,

0’

. . . anI 0 0 ..* 0

u2 ... a, 0 a.0 0

a1 a2 --* anI *** 0

0’ ao* a,’ u2 .. 1 a,

b,, 0 ... 0

b,, ... 0

b; b, . . . * i,,

A Note on Resultants 305

THEOREM 2.1. Necessary and sufficient conditions for f(x) and g(x) to have a common divisor of degree d, but none of higher degree, are R = 0, R,=O,..., R&l= 0, R, # 0, where R, is the determinant derived from R by deleting the last K rows of a’s, the lust k rows of b’s, and the lust 2k columns.

The second classical theorem is Kronecker’s method of elimination. (See [5].) Let f, be a polynomial with nonvanishing leading coefficient. Let

f 2,. . . , f, be some other polynomials. Form the resultant of fi and va fi + . . . + v, f,, where va,. . . , v, are indeterminants. Then a necessary and sufficient condition for all f; to have a common root is that all coefficients of the monomials in v’s shall vanish. The extension to larger numbers of common roots is clear by applying the first theorem. In this note we shall explicitly exhibit these coefficients. This is possible by elementary properties of determinants.

3. NOTATION

Let a = (al ,..., a,), b = (b, ,..., b,), etc. denote row vectors. The nota- tion (a, b,. . .) will denote the matrix block formed by placing a, b,. . . in rows, each successive row being shifted one entry to the right, and the rest being filled in by zeros in a case where a row vector does not have maximum dimension. The first row of each block is started in column one. Thus the classical resultant of two polynomials f( XT) and g(x) given earlier consists of two blocks: one (a,a,..., a) of dimension n, and one (b, b,. . . , b) of dimen- sion m. Let

f,(x) = aOrn + alxnpl + . . . + a,, a,#O,

A(x) = s bijXj, i = 2,..., h. j=l

Let m be the maximum of the degrees of the fi, i = 2,. . . , h. Make a selection Z of n b-rows, possibly with repetitions. Let

denote the determinant of the matrix formed from the two blocks A and B, where A=(a,a,..., a), of dimension m, and B is a block from the selection

306

I of n brows. Let

DAVID WEINBERG AND CLYDE MARTIN

where the sum extends over all permutations of the brows for a given selection 1. Let D,, denote the sum of determinants formed by deleting the last k rows of a ‘s. the last k rows of b’s, and the last 2k columns from each term of D,. We are now ready to state our

THEOREM 3.1. Necessary and suficient conditions for all J(x), i = 1 , . . . , h, to have a common divisor of degree d, but none of higher degree, are

D, = 0, D,l = 0,. . . , Dlcd+ = 0, Did # 0, where J ranges over all selections of n brows, and I is some particulur selection of n b-rows.

4. COMPLEX INVARIANT THEORY OF POLYNOMIALS

Given a polynomial

f(x) = a& + a$“-’ + . . . + a,, (1)

the question is: How can you tell the multiplicities of all the roots by looking at the coefficients? In other words, are there some polynomials in the coefficients of f(r) whose vanishing or nonvanishing will determine the multiplicities of all the roots?

The invariant theory (even the real invariant theory) of cubits and quartics is covered in many textbooks. See, for example, [3] or [4]. Perhaps the most familiar invariant of a polynomial is its discriminant, which, up to a constant factor, amounts to the resultant of f and f’. It vanishes if and only if f has a multiple root. A root common to f, f', f", . . . , f k- ' is a root of multiplicity at least k. Thus, by finding for each k the number of roots common to f and its first k derivatives, one can determine the multiplicities of all the roots. It is now clear that by applying the theorem in the previous section, one can explicitly exhibit the required invariants, expressed as sums of determinants formed from the coefficients of f.

EXAMPLE 4.1 (Cubits).

f(x) = a,x3+ a,x2+ a,r + u3, a,fO.

A Note on Resultants 307

LA

D,=

D2 =

Then we have

a0 (11 a2 a3 0

0 a0 a1 a2 a3

3u, 2u, u2 0 0

0 3u, 2u, u2 0

0 0 3u, 2u, u2

a0 a1 a2 3u, 2a, a2

0 3u, 2u,

Root multiplicities Conditions

3 distinct D,#O

1 double, 1 single D,=O, D,#O

1 triple D,=O, D,=O

EXAMPLE 4.2 (Quarks).

f(x)=uox4+u,r3+a2x2+u,x+u4, u,#O. (2)

We now refer to the notation established earlier, before the statement of the main theorem. Let

and let

f=bo, ai, a2~a3, a,>,

f = (4ao&,,2a2, a,),

f’= (12uo,6a,,2u2),

308

the discriminant. Then

DAVID WEINBERG AND CLYDE MARTIN

D,=

D3 =

a0 al a2 a3 a4 0 a0 a1 a2 a3

4a, 3a, 2a, a3 ,

0 4a, 3a, 2a, a3 0 0 4a, 3a, 2a,

a0 a1 a2

4a, 3a, 2a, ,

0 4a, 3a,

D,=l(f,f,f);(f’,f’,f,f”)(+l(f,f,f);(f,f,f’,f’)l

De= c l(f,f,f);(f,f’,f”,f’)l, KY, 7’)

where the sum extends over all six permutations of f’ and f’ in the second block, and

D,= c I(f,f,f);(f’,f”,f”,f’)I, o(fJ-9

where the sum extends over all four permutations of f’ and f’ in the second block. We have

Root multiplicities Conditions

4 distinct D,#O 2 single, 1 double D,=O, D,#O

2 doubles D,=O, D,=O, D,#O, D,#O forsome i=4,...,7

1 single, 1 triple D,#O, Di=O, i=1,2,4 ,..., 7 1 quadruple Di=O, i=1,2,3

A Note on Resultants 309

REFERENCES 1 S. Bamett, Greatest common divisors from generalized Sylvester resultant matrices,

Linear and Multilinear Algebra 8:271-279 (1980). 2 W. Burnside and A. Panton, The Theory of Equations, Vol. 1, Hodges, Figgis, and

Co., Dublin, 1928. 3 L. E. Dickson, EZementay Theoy of Equations, Wiley, New York, 1917. 4 C. Durell and A. Robson, Advanced Algebra, Vol. II, G. Bell and Sons, London,

1937. 5 B. L. Van der Waerden, Modern Algebra, Vol. II, Ungar, New York, 1950. 6 A. Vardulakis and P. Stoyle, Generalized resultant theorem, J. Inst. Math. Appl.

2233-335 (1978).