In 1853 Sylvester stated and proved an elegant formula that expresses the
polynomial subresultants in terms of the roots of the input polynomials.
Sylvester's formula was also recently proved by Lascoux and Pragacz by using
multi-Schur functions and divided differences. In this paper, we provide an
elementary proof that uses only basic properties of matrix multiplication and
Vandermonde determinants.Comment: 9 pages, no figures, simpler proof of the main results thanks to
  useful comments made by the referees. To appear in Journal of Symbolic
  Computatio

D'Andrea, Carlos

Hong, Hoon

Krick, Teresa

Szanto, Agnes

Journal of Symbolic Computation

English

arXiv

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.Fil: D'Andrea, Carlos. Universidad de Barcelona; EspañaFil: Hong, Hoon. North Carolina State University; Estados UnidosFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados Unido

Krick, Teresa Elena Genoveva

CONICET Digital

An elementary proof of Sylvester's double sums for subresultants

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants. © 2006 Elsevier Ltd. All rights reserved.Fil:D'Andrea, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Krick, T. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

D'Andrea, C.

Hong, H.

Krick, T.

Szanto, A.

Repositorio Digital Institiucional Universidad de Buenos Aires

Journal of Symbolic Computation 42 (2007) 290–297www.elsevier.com/locate/jscAn elementary proof of Sylvester’s double sums forsubresultantsCarlos D’Andreaa, Hoon Hongb, Teresa Krickc, Agnes Szantob,∗a Department d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les CortsCatalanes, 585; 08007, Spainb Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USAc Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, 1428 Buenos Aires, ArgentinaReceived 20 September 2005; accepted 14 September 2006Available online 13 November 2006AbstractIn 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants interms of the roots of the input polynomials. Sylvester’s formula was also recently proved by Lascoux andPragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proofthat uses only basic properties of matrix multiplication and Vandermonde determinants.c© 2006 Elsevier Ltd. All rights reserved.Keywords: Subresultants; Double-sum formula; Vandermonde determinant1. IntroductionSubresultants play a fundamental role in Computer Algebra and Computational AlgebraicGeometry (for instance, see Collins (1967), Brown and Traub (1971), Collins (1975), Gonzalez-Vega et al. (1989), Renegar (1992), Apéry and Jouanolou (2005), Gonzalez-Vega (1996),Lombardi et al. (2000) and Hong (2001)). Sylvester (1853) stated and proved an elegant formulathat expresses the polynomial subresultants of two polynomials in terms of their roots, the so-called double-sum formula. This identity was proved also by Lascoux and Pragacz (2003), byusing the theory of multi-Schur functions and divided differences.∗ Corresponding author. Tel.: +1 919 622 7454; fax: +1 919 513 7336.E-mail addresses: cdandrea@ub.edu (C. D’Andrea), hong@math.ncsu.edu (H. Hong), krick@dm.uba.ar (T. Krick),aszanto@ncsu.edu (A. Szanto).0747-7171/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jsc.2006.09.003C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297 291In this paper we provide a new and elementary proof that uses only the basic propertiesof matrix multiplication and Vandermonde determinants. As apparent in our proof, Sylvester’sdouble-sum formula is only one simple step further from a particular case, the so-called single-sum formula. Such connection between the single and the double-sum formulae was originallythought to be unlikely, as remarked in page 691 of Lascoux and Pragacz (2003). There havebeen various proofs for the single-sum formula (Borchardt, 1860; Chardin, 1990; Apéry andJouanolou, 2005; Hong, 1999; Diaz-Toca and Gonzalez-Vega, 2004).The matrix multiplication technique, presented in this paper, has proven to be quite powerfulin that it is easily generalizable to multivariate polynomials: similar techniques were successfullyapplied to obtain expressions for multivariate subresultants in roots in D’Andrea et al. (2006),and the generalization of Sylvester’s single and double-sum formulae to the multivariate case isthe subject of ongoing research.2. Review of Sylvester’s double sum for subresultantsLet f = am xm + · · · + a0 and g = bn xn + · · · + b0, be two polynomials with coefficientsin a commutative ring. The dth subresultant polynomial Sresd( f, g) is defined for 0 ≤ d <min{m, n}, and if m 6= n holds, also for d = min{m, n}, as the following determinant:Sresd( f, g) := detm+n−2dam · · · · · · ad+1−(n−d−1) xn−d−1 f (x). . ....... n−dam · · · ad+1 f (x)bn · · · · · · bd+1−(m−d−1) xm−d−1g(x). . ....... m−dbn · · · bd+1 g(x)(1)where a` = b` = 0 for ` < 0.By developing this determinant by the last column, it is clear that Sresd( f, g) is a polynomialcombination of f and g. It is also a classic fact that Sresd( f, g) is a polynomial of degree boundedby d , since it coincides with the determinant of the matrix obtained by replacing the last columnCm+n−2d withC ′m+n−2d := Cm+n−2d − xd+1Cm+n−2d−1 − · · · − xm+n−d−1C1.Now, let A = (. . . , α, . . .) and B = (. . . , β, . . .) be finite lists (ordered sets) of distinctindeterminates. Sylvester (1853) introduced for 0 ≤ p ≤ |A|, 0 ≤ q ≤ |B| the followingdouble-sum expression in A and B:Sylvp,q(A, B; x) :=∑A′⊂A, B′⊂B|A′|=p, |B′|=qR(x, A′) R(x, B ′)R(A′, B ′) R(A\A′, B\B ′)R(A′, A\A′) R(B ′, B\B ′),whereR(X, Y ) :=∏x∈X,y∈Y(x − y), R(x, Y ) :=∏y∈Y(x − y).292 C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297Sylvester (1853) gave the following elegant formula that expresses the subresultants in termsof the double sum, that is, in terms of the roots of f and g.Theorem 1 (Sylvester’s Double-sum Formula). Let f, g be the monic polynomialsf =∏α∈A(x − α), g =∏β∈B(x − β) ∈ Z[α ∈ A, β ∈ B][x],where |A| = m and |B| = n. Let p, q ≥ 0 be such that d := p + q < min{m, n} ord ≤ min{m, n} if m 6= n holds. ThenSresd( f, g) =(−1)p(m−d)(dp) Sylvp,q(A, B; x).When p = d and q = 0, the above expression immediately simplifies to the single-sum formula:Sresd( f, g) =∑A′⊂A|A′|=dR(x, A′)R(A\A′, B)R(A\A′, A′). (2)Complete proofs of Sylvester’s double sum can be found in Sylvester (1853) and Lascouxand Pragacz (2003), while the single-sum formula has various proofs (Borchardt, 1860; Chardin,1990; Apéry and Jouanolou, 2005; Hong, 1999; Diaz-Toca and Gonzalez-Vega, 2004). Here wepresent in Section 4 an alternative elementary proof for both results.3. NotationsWe recall that 0 ≤ d < min{m, n} or d ≤ min{m, n} if m 6= n holds. We let M f and Mgdenote the following matrices:M f :=m+n−da0 . . . am. . .. . . n−da0 . . . am, Mg :=m+n−db0 . . . bn. . .. . . m−db0 . . . bn.We now defineSd :=m+n−dMt−x dM f n−dMg m−dwhere Mt−x :=m+n−d−x 1 0 . . . . . . 0. . .. . .. . .... d−x 1 0 . . . 0.Finally, we define for a polynomial p(t) and two lists, Γ := (γ1, . . . , γu) of scalars andE := (e1, . . . , ev) of non-negative integers, the (not-necessarily square) matrix of size v × u:C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297 293〈p(t),Γ 〉E :=uγe11 p(γ1) . . . γe1u p(γu)...... vγev1 p(γ1) . . . γevu p(γu).For instance, under this notation, if we take E := (0, . . . , u − 1), we have the following equalityfor the Vandermonde determinant V(Γ ) associated to Γ :V(Γ ) := |(γ i−1j )1≤i, j≤u | = |〈1,Γ 〉E |.When E is of the form E = (0, . . . , v − 1), we directly write 〈p(t),Γ 〉v .We mention the following useful equalities that hold since m + n − d ≥ max(m, n):M f · 〈1,Γ 〉m+n−d = 〈 f (t),Γ 〉n−dMg · 〈1,Γ 〉m+n−d = 〈g(t),Γ 〉m−dMt−x · 〈1,Γ 〉m+n−d = 〈t − x,Γ 〉d .4. The proofThe proof is divided into a series of lemmas which are interesting on their own. For an easierunderstanding, we recommend not to pay attention to signs in a first approach.Lemma 1. Under the previous assumptions and notations, we haveSresd( f, g) = (−1)d+(n−d)(m−d)|Sd |.Proof. We denote by Ci the i th column of the matrix Sd and we replace its first column C1 byC ′1 := C1 + xC2 + · · · + xm+n−d−1Cm+n−d . This operation does not change the determinant ofthis matrix, andC ′1 =0... d0f (x)... n−dxn−d−1 f (x)g(x)... m−dxm−d−1g(x).We now perform a Laplace expansion of the determinant of the new matrix over the first d rows,and we observe that only one block survives, which corresponds to columns 2 to d + 1 of Mt−x .Moreover, this block is lower triangular with diagonal entries 1. Thus294 C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297|Sd | = (−1)d detm+n−2df (x) ad+1 . . . am....... . . n−dxn−d−1 f (x) ad+1−(n−d−1) . . . . . . amg(x) bd+1 . . . bn....... . . m−dxm−d−1g(x) bd+1−(m−d−1) . . . . . . bn= (−1)d+(n−d)(m−d) Sresd ( f, g),since the matrix in the right-hand side above is the matrix of (1) viewed backward. For simplicity, from now on, we assume f and g to be the monic polynomials f =∏α∈A(x − α), g =∏β∈B(x − β) where A and B are lists with |A| = m and |B| = n. (Aspointed out by a referee, under this assumption one has in the language of multi-Schur functions:|Sd | = S1d ;(m−d)n−d ;0m−d (−x, −A, −B) (see Lascoux and Pragacz (2003)).)The lemmas below generalize in an obvious manner to non-monic polynomials. The first onecorresponds to Th. 3 in Hong (1999). We prove it here with a different technique that followsfrom Lemma 1.Lemma 2 (Hong’s Subresultant in Roots (Hong, 1999, Th. 3.1)). Under the previous notations,we haveSresd( f, g)V (A) = detm〈x − t, A〉d d〈g(t), A〉m−d m−d.Proof. We note that |Sd | V(A) is the determinant of the following product of matrices:m+n−dd Mt−xn−d M fm−d Mgm n−d0 m〈1, A〉m+n−dIn−d n−d=m n−d〈t − x, A〉d ∗ d0 M ′f n−d〈g(t), A〉m−d ∗ m−d,since 〈 f (t), A〉n−d =[αi−1j f (α j )]= [0].By permuting the rows of the second block with those of the third, we obtainSresd( f, g)V(A) = (−1)d+(m−d)(n−d)|Sd |V(A)= (−1)d det〈t − x, A〉d〈g(t), A〉m−d|M ′f |= det〈x − t, A〉d〈g(t), A〉m−d,since M ′f is a lower triangular matrix with diagonal entries am = 1. C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297 295Let us remark here that the Poisson product formula Res( f, g) =∏α∈A g(α) is a directconsequence of the previous lemma for the case d = 0.For S ⊆ T finite lists, let sg (S, T ) := (−1)σ where σ is the number of transpositions neededto take T to S ∪ (T \S). Here, “∪” stands for list concatenation and “\” means list subtraction.Lemma 3. Let P and Q be two disjoint sublists of E := (0, . . . , d − 1) that satisfy P ∪ Q = E,and let p := |P|, q := |Q|. ThenSresd ( f, g)V(A)V(B) = (−1)q+(m−d)n sg (P, E) detm n〈x − t, A〉P 0 p0 〈x − t, B〉Q q〈1, A〉m+n−d 〈1, B〉m+n−d m+n−d.(3)Proof. Recalling that V(B) = |〈1, B〉n|, we have by Lemma 2:Sresd ( f, g)V(A)V(B) = detm n〈x − t, A〉d 0 d〈g(t), A〉m−d 0 m−d〈1, A〉n 〈1, B〉n n= (−1)(m−d)n detm n〈x − t, A〉d 0 d〈1, A〉n 〈1, B〉n n〈g(t), A〉m−d 0 m−d= (−1)(m−d)n detd n m−dd Id 0 0n 0 In 0m−d 0 Mgm n〈x − t, A〉d 0 d〈1, A〉m+n−d 〈1, B〉m+n−d m+n−d ,since Mg · 〈1, B〉m+n−d = 〈g(t), B〉m−d = [0]. Now, since the first matrix is lower triangularwith diagonal entries 1, we haveSresd( f, g)V(A)V(B) = (−1)(m−d)n detm n〈x − t, A〉d 0 d〈1, A〉m+n−d 〈1, B〉m+n−d m+n−d. (4)Finally, recalling that 〈x − t, A〉d =(αi−1j x − αij)1≤i≤d,1≤ j≤mand 〈1, A〉m+n−d =(αi−1j)1≤i≤m+n−d,1≤ j≤m, the obvious subtractions and permutations of rows yieldSresd ( f, g)V(A)V(B) = (−1)(m−d)n sg (P, E) detm n〈x − t, A〉P 0 p0 −〈x − t, B〉Q q〈1, A〉m+n−d 〈1, B〉m+n−d m+n−d.296 C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297The lemma follows by moving (−1)q out of the determinant. We will also need the following observation:Observation 1. Let Γ := (γ1, . . . , γd). Then|〈x − t,Γ 〉d | = R(x,Γ )|〈1,Γ 〉d |. (5)Proof. The claim follows from x − γ1 . . . x − γd... ...γ d−11 x − γd1 . . . γd−1d x − γdd = 1 . . . 1... ...γ d−11 . . . γd−1dx − γ1 . . .x − γd .4.1. Proof of Theorem 1For any P and Q disjoint sublists of E := (0, . . . , d − 1) that satisfy P ∪ Q = E , with|P| = p and |Q| = q , a Laplace expansion over the first d rows in Identity (3) gives thatSresd( f, g)V (A)V (B) equals(−1)σ sg(P, E)∑A′⊂A, B′⊂B|A′|=p,|B′|=qsg(A′, A) sg(B ′, B) · |〈x − t, A′〉P | · |〈x − t, B ′〉Q |·V(A\A′ ∪ B\B ′)where σ := q + (m −d)n + (m − p)q ≡ (m −d)(n −q) (mod 2). Adding over all such choicesof P ⊂ E with |P| = p, we deduce that Sresd( f, g)V (A)V (B) equals1(dp) ∑P(−1)σ sg (P, E)∑A′,B′sg(A′, A) sg(B ′, B) · |〈x − t, A′〉P | · |〈x − t, B ′〉Q |·V(A\A′ ∪ B\B ′)=(−1)σ(dp) ∑A′,B′sg(A′, A)sg(B ′, B)V(A\A′ ∪ B\B ′)·(∑Psg (P, E) |〈x − t, A′〉P | · |〈x − t, B ′〉Q |).We observe now that, by another Laplace expansion and Identity (5),∑Psg (P, E) |〈x − t, A′〉P | · |〈x − t, B ′〉Q | = |〈x − t, A′ ∪ B ′〉d |= R(x, A′)R(x, B ′)|〈1, A′ ∪ B ′〉d |.Recalling that |〈1, A′ ∪ B ′〉d | = V(A′ ∪ B ′), this givesSresd( f, g)=(−1)σ(dp) ∑A′,B′R(x, A′)R(x, B ′)sg(A′, A) sg(B ′, B)V(A\A′ ∪ B\B ′)V(A′ ∪ B ′)V(A)V(B)=(−1)σ (−1)τ(dp) ∑A′,B′R(x, A′)R(x, B ′)R(A′, B ′)R(A\A′, B\B ′)R(A′, A\A′)R(B ′, B \ B ′),C. D’Andrea et al. / Journal of Symbolic Computation 42 (2007) 290–297 297where τ = (m − p)(n − q) + pq − (m − p)p − (n − q)q = (m − d)(n − d) since for any finitelists X, Y , one has V(X ∪ Y ) = V(X)V(Y )R(Y, X) = (−1)|X |·|Y |V(X)V(Y )R(X, Y ). The claimfollows now from the fact that (m − d)(n − q) + (m − d)(n − d) ≡ (m − d)p (mod 2). As a final remark, we mention that if in the previous proof we start with a Laplace expansionover the first d rows in Identity (4) instead of Identity (3), we obtain in the same mannerSylvester’s single sum formulation (2).AcknowledgementsWe are grateful to the anonymous referees for their careful reading of our preliminarymanuscript and their very precise indications to improve our presentation.HH’s research was supported by NSF grant CCR-0097976, TK’s by CONICET PIP 2461/01and UBACYT X-112 grants and AS’s by NSF grants CCR-0306406 and CCR-0347506.ReferencesApéry, F., Jouanolou, J.P., 2005. Résultant et sous-résultants: le cas d’une variable, Cours DESS 1995–1996.Borchardt, C.W., 1860. Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über DerenAnwendung. Math. Abh. der Akademie der Wissenschaften zu Berlin, pp. 1–20.Brown, W.S., Traub, J.F., 1971. 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In: Texts and Monographs inSymbolic Computation, Springer-Verlag.Gonzalez-Vega, L., Lombardi, H., Recio, T., Roy, M.-F., 1989. Sturm–Habicht sequences. In: Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, July 1989, pp. 136–146.Hong, H., 1999. Subresultants in roots. Technical Report. Department of Mathematics, North Carolina State University.Hong, H., 2001. Ore subresultant coefficients in solutions. Journal of Applicable Algebra in Engineering,Communication, and Computing 12 (5), 421–428.Lascoux, A., Pragacz, P., 2003. Double Sylvester sums for euclidean division, multi-Schur functions. Journal of SymbolicComputation 35, 689–710.Lombardi, H., Roy, M.-F., El Din, M.S., 2000. New structure theorem for subresultants. Journal of SymbolicComputation 29, 663–689.Renegar, J., 1992. On the computational complexity and geometry of the first-order theory of the reals. Journal ofSymbolic Computation 13 (3), 255–352.Sylvester, J.J., 1853. On a theory of syzygetic relations of two rational integral functions, comprising an applicationto the theory of Sturm’s function and that of the greatest algebraical common measure. Trans. Roy. Soc. London.Reprinted in: The Collected Mathematical Papers of James Joseph Sylvester, vol. 1, Chelsea Publ., New York, 1973,pp. 429–586.

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An Elementary Proof of Sylvester's Double Sums for Subresultants

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