Simple Proofs of Some Theorems in Resultant Theory

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 16 (2015), No 1, pp. 205-211 DOI: 10.18514/MMN.2015.1369

Simple proofs of some theorems in resultant theory

Amir Hashemi and Farzad Mirzavand Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 16 (2015), No. 1, pp. 205–211

SIMPLE PROOFS OF SOME THEOREMS IN RESULTANT THEORY

AMIR HASHEMI AND FARZAD MIRZAVAND Received 10 October, 2014

Abstract. In this note, using elementary results from algebraic geometry, we give new and simple proofs of some known theorems in resultant theory.

2010 Mathematics Subject Classiﬁcation: 13P15 Keywords: homogeneous polynomials, multivariate resultants, three ternary quadrics system

1. INTRODUCTION The multivariate resultant is a fundamental tool of computational algebraic geometry which was introduced by F. S. Macaulay [4] in 1902 after earlier work due to Euler, Sylvester and Cayley. The resultant of a polynomial system has many im- portant properties for the geometry of the variety that the system defines. Indeed, the resultant is an algebraic condition in terms of the coefficients of a given system of polynomials which is satisfied if and only if the system has a common solution. More l precisely, suppose that K is an algebraically closed field. Let f a0 al x m D C C and g b0 bmx be two univariate polynomials where ai ’s and bi ’s belong D C C to K and a ;bm 0. The Sylvester matrix of f;g is defined as l ¤ 2 a0 b0 3 6 a1 a0 b1 b0 7 6 7 6 :: :: 7 6 a2 a1 : b2 b1 : 7 6 : : : : 7 6 : :: : :: 7 6 : a2 a0 : b2 b0 7 S.f;g/ 6 : : : : 7: D 6 a : :: a b : :: b 7 6 l 1 m 1 7 6 a a b b 7 6 l 2 m 2 7 6 : : : : 7 4 :: : :: : 5 al bm The univariate resultant of f and g is the determinant of S.f;g/ and is denoted by Res.f;g/. It is well known that Res.f;g/ 0 if and only if f and g have a D c 2015 Miskolc University Press

206 AMIR HASHEMI AND FARZAD MIRZAVAND nontrivial common root (see [1, Chapter 3, Proposition 1.7]). A generalization of this result can be used to decide whether a system of n 1 homogeneous equa- tions in n 1 variables has a solution or not. ThroughoutC this note we let R C D KŒx0;:::;xn. To state the next theorem, we need to introduce some notation. Let F0;:::;Fn R be n 1 homogeneous polynomials of degrees d0;:::;dn. So, we can write F2 as P C c x˛ where some of the c ’s may be zero. For each i ˛ di i;˛ i;˛ j jD possible pair of indices i;˛, we introduce a new variable ui;˛. Let us associate a polynomial P ZŒu to the universal homogeneous polynomials P u x˛ i;˛ ˛ di i;˛ for i 0;:::;n2. Then, when the polynomial P for a particular collectionj jD of poly- nomialsD P c x˛ for i 0;:::;n is considered, for each i and ˛, the variable ˛ di i;˛ j jD D ui;˛ is substituted by ci;˛ into P .

Theorem 1 ([1, Chapter 3, Theorem 2.3]). Let us ﬁx the degrees d0;:::;dn. Then there is a unique polynomial Res ZŒui;˛ which has the following properties: 2 .1/ If F0;:::;Fn R are homogeneous of degrees, respectively d0;:::;dn, then 2 the system F0 Fn 0 have a nontrivial solution over K if and only D D D if Res.F0;:::;Fn/ 0, d0 dn D .2/ Res.x ;:::;xn / 1, 0 D .3/ Res is irreducible as a polynomial in KŒui;˛.

If we ﬁx the degrees d0;:::;dn, then the polynomial Res Res ZŒui;˛ D d0;:::;dn 2 is called the resultant polynomial corresponding to d0;:::;dn. Further, the constant Res.F0;:::;Fn/ K is called the resultant of F0;:::;Fn. For more details, we refer the reader to the books2 [1,2,5]. The resultant has many algebraic properties that make it a convenient tool in constructive algebra. In this note, we consider the symmetry and multiplicativity property of resultant.

Theorem 2. Suppose that F0;:::;Fn R are homogeneous of degrees, respect- 2 ively d0;:::;dn. .a/ If i < j then

d0 dn Res.F0;:::;Fi ;:::;Fj ;:::;Fn/ . 1/ Res.F0;:::;Fj ;:::;Fi ;:::;Fn/: D .b/ If Fj F F is a product of homogeneous polynomials of degrees d and D j0 j00 j0 dj00 then

Res.F0;:::;Fn/ Res.F0;:::;F ;:::;Fn/Res.F0;:::;F ;:::;Fn/: D j0 j00 In 1991, Jouanolou [3, Section 5], proved this theorem, however, in Section2 we give a new and elementary proof for it. Further, in Section3, we use some effective elementary algebraic geometry results to prove the necessary and sufﬁcient condi- tions for the satisﬁability of the “three ternary quadrics” system. For the classical proof of this result, we refer the reader to ([6, Art. 90] and [1, page 88]). SIMPLE PROOFS OF SOME THEOREMS IN RESULTANT THEORY 207

2. THEPROOFOF THEOREM 2 In order to prove this theorem, we need some properties of ideals and varieties. Let R KŒx0;:::;xn be a polynomial ring over an algebraically closed field K and D I R a homogeneous ideal. Further, let f1;:::;f be polynomials (not necessarily k homogeneous) in R. Then, the variety defined by f1;:::;fk is the set n 1 V.f1;:::;f / .a0;:::;an/ K fi .a0;:::;an/ 0 for all i : k D f 2 C j D g n 1 A subset V K is called a variety if there exist f1;:::;f R so that V C k 2 D V.f1;:::;fk/. Further, the ideal of a variety V is defined to be I.V / f R f .a0;:::;an/ 0 for all .a0;:::;an/ V . D f 2 j D 2 g Theorem 3 (Hilbert’s Nullstellensatz). Suppose that I f1;:::;f is the ideal D h ki generated by the fi ’s. Then, I.V.f1;:::;f // pI . k D For more details on this topic we refer the reader to [1,2].

Theorem 4 ([1, Theorem 3.1, page 95]). Let us ﬁx the degrees d0;:::;dn, then the Pn polynomial Resd0;:::;dn has the total degree j 0 d0 dj 1dj 1 dn. D C We continue with some notation. Let F R be a homogeneous polynomial. Then, 2 we denote the polynomial F .x0;:::;xn 1;0/ and F .x0;:::;xn 1;1/ by F .x0;:::;xn 1/ and f .x0;:::;xn 1/, respectively. N Theorem 5 ([1, Theorem 3.4, page 96]). If Res.F ;:::;F / 0, then we have N0 Nn 1 dn ¤ Res.F0;:::;Fn/ Res.F0;:::;Fn 1/ :det.mfn / where mfn is the linear multiplic- D N N ation map by fn on KŒx0;:::;xn 1= f0;:::;fn 1 . h i Proof. of Theorem2: .a/ Let us denote by Ri;j and Rj;i the polynomials Res.F0;:::;Fn/ and Res.F0;:::;Fj ;:::;Fi ;:::;Fn/, respectively. By Theorem1, if Rj;i vanishes, then

F0 Fj Fi Fn 0 D D D D D D D has a nontrivial solution. On the other hand, every nontrivial solution of this system is a nontrivial solution of F0 Fn 0. Hence Ri;j vanishes on the set V.Rj;i /. This shows that D D D q Ri;j I.V.Rj;i // Rj;i : 2 D h i However, Rj;i is irreducible, which implies that Ri;j Rj;i . By Theorem4, Ri;j 2 h i and Rj;i have the same degree. This follows that there exists a constant c so that d0 dn Ri;j cRj;i . We shall prove that c . 1/ . Since c is constant, it’s enough to showD that D d Res.xd0 ;:::;xdn / . 1/d0 dn Res.xd0 ;:::;x j ;:::;xdi ;:::;xdn /: 0 n D 0 j i n In doing so, we will use induction on the number of polynomials. For n 2, the claim is implied by the main properties of determinants. Now assume that theD claim holds 208 AMIR HASHEMI AND FARZAD MIRZAVAND

d0 dn for n polynomials of degrees d0;:::;dn 1. By applying Theorem5 on x0 ;:::;xn d0 dj di dn and x0 ;:::;xj ;:::;xi ;:::;xn , it follows respectively that d0 dn Res.x0 ;:::;xn / det.m dn / d d xn 0 n 1 dn D Res.x0 ;:::;xn 1 / d0 dj di dn Res.x0 ;:::;xj ;:::;xi ;:::;xn / d det.m n / d d d d : xn 0 j i n 1 dn D Res.x0 ;:::;xj ;:::;xi ;:::;xn 1 / Thus, we can write

d dj d d d0 dn Res.x 0 ;:::;x ;:::;x i ;:::;x n / Res.x0 ;:::;xn / 0 j i n d0 dn 1 d dn D d0 j di dn 1 dn Res.x0 ;:::;xn 1 / Res.x0 ;:::;xj ;:::;xi ;:::;xn 1 / and therefore from the induction hypothesis we can conclude that

d0 dn Res.x0 ;:::;xn / d d d d d 0 n 1 dn 0 j i dn Res.x ;:::;x / Res.x ;:::;x ;:::;x ;:::;xn / 0 n 1 0 j i d d d d 0 j i n 1 dn D Res.x0 ;:::;xj ;:::;xi ;:::;xn 1 /

d d d d d d d d d0 dn 0 j i n 1 dn 0 j i n . 1/ Res.x ;:::;x ;:::;x ;:::;x / Res.x ;:::;x ;:::;x ;:::;xn / 0 j i n 1 0 j i d d d d 0 j i n 1 dn D Res.x0 ;:::;xj ;:::;xi ;:::;xn 1 /

d0 dn d0 dj di dn . 1/ Res.x ;:::;x ;:::;x ;:::;xn / D 0 j i which proves the assertion. To prove .b/, let us denote by R;R0;R00 the polynomials Res.F0;:::;Fj ;:::;Fn/;Res.F0;:::;Fj0;:::;Fn/;Res.F0;:::;Fj00;:::;Fn/. We shall show that R R R . Note that R may be considered as a polynomial in the coef- D 0 00 ficients of F0;:::;Fj0;Fj00;:::;Fn: By Theorem1, if either R0 or R00 vanishes, then either the system F0 F Fn 0 or the system F0 F D D j0 D D D D D j00 D D Fn 0 has a nontrivial solution. Every nontrivial solution of these systems is a non- D trivial solution of the system F0 Fn 0: Note that we consider R;R and R D D D 0 00 as the polynomials in the coefficients of polynomials F0;:::;Fj0;Fj00;:::;Fn;. Then due to the irreducibility of R0 and R00, we have p p R I.V.R // R R ;R I.V.R // R R : 2 0 D h 0i D h 0i 2 00 D h 00i D h 00i This proves that R R and R R: It should be noted that R and R have degrees 0 j 00 j 0 0 and d0 dj 1dj 1 dn in the coefficients of polynomial Fj00, respectively. Also, C R0 and R have degree d0 dj 1dj 1 dn in the coefficients of polynomial Fj0. C Thus, there exists a polynomial g such that R gR where g has degree 0 and D 0 d0 dj 1dj 1 dn in the coefficients of Fj0 and Fj00, respectively. From R R00 C 2 h i it implies that R gR : On the other hand, R is irreducible, which implies that R 00 j 0 00 00 divides either g or R0: Since R0 and R00 have degrees 0 and d0 dj 1dj 1 dn in C SIMPLE PROOFS OF SOME THEOREMS IN RESULTANT THEORY 209 the coefficients of F ; it follows that R g: Since R and g have the same degree j00 00 j 00 in the coefficients of polynomial Fj00 there exists a polynomial c of degree 0 in the coefficients of F such that g cR . Thus, we conclude that R cR R : Note that j00 D 00 D 0 00 R and R0R00 have the same total degree. Therefore c is constant. From the equality d d d0 j0 j00 dn F0 x ;:::;F x ;F x ;:::;Fn x ; D 0 j0 D j j00 D j D n we see that R R R 1 and thus c 1; which implies that R R R and this D 0 D 00 D D D 0 00 ends the proof.

3. THREETERNARYQUADRICSSYSTEM In this section we consider the classical system of three ternary quadrics. This is the following system 2 2 2 F0 c01x c02y c03´ c04xy c05x´ c06y´ 0 D 2 C 2 C 2 C C C D F1 c11x c12y c13´ c14xy c15x´ c16y´ 0 D 2 C 2 C 2 C C C D F2 c21x c22y c23´ c24xy c25x´ c26y´ 0 D C C C C C D where the cij ’s are parameters. By Theorem1, Res .F0;F1;F2/ vanishes exactly when this system has a nontrivial solution in x;y and ´. However, Res.F0;f1;F2/ is a large polynomial in 18 variables with 21894 terms (see [1, page 88]). The aim of this section is to provide a new and simple proof for a compact representation of Res.F0;F1;F2/. For the original proof, we refer to [6, Art. 90]. Let us denote by J the Jacobian determinant of F0;F1;F2 w.r.t. the variables x;y and ´. Then, the partial derivatives of J are quadratic and hence we can write @J 2 2 2 @x b01x b02y b03´ b04xy b05x´ b06y´ @J D 2 C 2 C 2 C C C @y b11x b12y b13´ b14xy b15x´ b16y´ @J D 2 C 2 C 2 C C C b21x b22y b23´ b24xy b25x´ b26y´: @´ D C C C C C Proposition 1. 2 3 c01 c02 c03 c04 c05 c06 6 c11 c12 c13 c14 c15 c16 7 6 7 1 6 c21 c22 c23 c24 c25 c26 7 Res.F0;F1;F2/ det6 7: D 512 6 b01 b02 b03 b04 b05 b06 7 6 7 4 b11 b12 b13 b14 b15 b16 5 b21 b22 b23 b24 b25 b26 Proof. We refer to the right hand side matrix as A. If we regard the monomials 2 2 2 @J @J @J x ;y ;´ ;xy;x´;y´ as unknowns, then F0;F1;F2; @x ; @y ; @´ are linear and @J @J @J Res1;1;1;1;1;1.F0;F1;F2; ; ; / det.A/: @x @y @´ D 210 AMIR HASHEMI AND FARZAD MIRZAVAND

Note that each bij is a cubic polynomial in the cij ’s. Hence the polynomial @J @J @J Res.F0;F1;F2; @x ; @y ; @´ / has total degree 12 in c01;:::;c26: By Theorem1, the resultant vanishes iff @J @J @J F0 F1 F2 0 D D D @x D @y D @´ D has a nontrivial solution .x2;y2;´2;xy;x´;y´/. Also, it can be easily veriﬁed that every nontrivial solution of this system is a nontrivial solution of F0 F1 F2 0: D D D Hence .x;y;´/ is a nontrivial solution of F0 F1 F2 0. Then D D D Res2;2;2.F0;F1;F2/ 0: Thus Res2;2;2.F0;F1;F2/ vanishes on the set D @J @J @J V.Res.F0;F1;F2; ; ; //: @x @y @´

This means that Res2;2;2.F0;F1;F2/ belongs to the ideal of this variety, i.e. s @J @J @J Res2;2;2.F0;F1;F2/ Res.F0;F1;F2; ; ; / : 2 h @x @y @´ i

On the other hand, if we substitute every bij in the above matrix by the corresponding cubic polynomial in the cij ’s, using the function irreduc of MAPLE, we can see easily that det.A/ and therefore @J @J @J Res.F0;F1;F2; ; ; / @x @y @´ is irreducible in the cij ’s which yields that @J @J @J Res2;2;2.F0;F1;F2/ Res.F0;F1;F2; ; ; / : 2 h @x @y @´ i

By Theorem4, Res 2;2;2.F0;F1;F2/ has the total degree 12 in c01;:::;c26: Hence, for a c K; we have 2 @J @J @J Res2;2;2.F0;F1;F2/ c Res.F0;F1;F2; ; ; /: D @x @y @´ 2 2 2 But, for the special case; if we set .F0;F1;F2/ .x ;y ;´ /; we can conclude that D 2 1 0 0 0 0 0 3 6 0 1 0 0 0 0 7 6 7 2 2 2 6 0 0 1 0 0 0 7 Res.x ;y ;´ ;8y´;8x´;8xy/ det6 7 512: D 6 0 0 0 0 0 8 7 D 6 7 4 0 0 0 0 8 0 5 0 0 0 8 0 0

2 2 2 1 Finally, by Theorem1, we have Res 2;2;2.x ;y ;´ / 1 and so c . D D 512 SIMPLE PROOFS OF SOME THEOREMS IN RESULTANT THEORY 211

ACKNOWLEDGEMENTS The research of the ﬁrst author was supported in part by a grant from IPM (No. 93550420). The authors are grateful to the referee and Prof. David Cox for their useful comments and advices for the improvement of this note.

REFERENCES [1] D. Cox, J. Little, and D. O’Shea, Using algebraic geometry. 2nd ed., 2nd ed. New York, NY: Springer, 2005. [2] D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 3rd ed., 3rd ed. New York, NY: Springer, 2007. [3] J. P. Jouanolou, “Le formalisme du resultant.´ (The formalism of resultant).” Adv. Math., vol. 90, no. 2, pp. 117–263, 1991. [4] F. Macaulay, “Some formulae in eliminations.” Proc. Lond. Math. Soc., vol. 35, pp. 3–27, 1903. [5] B. Mishra, Algorithmic algebra. Berlin: Springer, 1993. [6] G. Salmon, “Lec¸ons d’algebre` superieure.´ Trad. de l’anglais par M. Bazin; augm. de notes par M. Hermite.” Paris, Gauthier-Villars. 1868 (1868)., 1868.

Authors’ addresses

Amir Hashemi Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395- 5746, Iran E-mail address: [email protected]

Farzad Mirzavand Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran E-mail address: [email protected]