Real Univariate Quintics

Abstract

For the general monic quintic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities together with the order of the real roots with respect to multiplicity.

Full text

arXiv:1901.03679v1 [math.AC] 11 Jan 2019
REAL UNIVARIATE QUINTICS
ELIAS GONZALEZ AND DAVID A. WEINBERG
Abstract. For the general monic quintic with real coefficients, polynomial
conditions on the coefficients are derived as directly and as simply as possible
from the Sturm sequence that will determine the real and complex root multiplicities
together with the order of the real roots with respect to multiplicity.
1. Quintics
1.1. Introduction. Consider the partition of the space of monic quintics, x5+
px4+qx3+rx2+sx +t, according to the following root configurations:
(1) 5 distinct real roots
(2) 3 distinct real roots and 2 distinct complex conjugate roots
(3) 1 real root and 4 distinct complex roots
(4) 1 double real root and 3 distinct real roots
(a) single root <double root <single root <single root
(b) double root <single root <single root <single root
(c) single root <single root <single root <double root
(d) single root <single root <double root <single root
(5) 1 double real root and 1 single real root and 2 distinct complex conjugate
roots
(a) single root <double root
(b) double root <single root
(6) 2 real double roots and 1 single real root
(a) single root <double root <double root
(b) double root <single root <double root
(c) double root <double root <single root
(7) 2 complex conjugate double roots and 1 single real root
(8) 1 triple root and 2 single real roots
(a) triple root <single root <single root
(b) single root <triple root <single root
(c) single root <single root <triple root
(9) 1 triple root and 2 complex conjugate roots
(10) 1 quadruple root and 1 single root
(a) quadruple root <single root
1

2 ELIAS GONZALEZ AND DAVID A. WEINBERG
(b) single root <quadruple root
(11) 1 triple root and 1 double root
(a) triple root <double root
(b) double root <triple root
(12) 1 quintuple root
We will find polynomial conditions on the coefficients p,q,r,s, and tthat will
determine to which of these classes the polynomial belongs. We will also determine
polynomial conditions on the coefficients that will determine the order of the real
roots with respect to multiplicity. (Thus, there will be three sets of results: two for
real polynomials and one for complex polynomials.) In a previous paper [3], the
authors gave the analogous results for cubics and quartics.
If c={c1, c2,...,cm}is a finite sequence of real numbers, then the number of
variations in sign of cis defined to be the number of i, 1 ≤i≤m−1 such that
cici+1 <0, after dropping the 0’s in c. The Sturm sequence for f(x) is defined to
be f0(x) = f(x), f1(x) = f′(x), f2(x), . . . , fs(x), where
f0(x) = q1(x)f1(x)−f2(x),degf2(x)<degf1(x)
.
.
.
fi−1(x) = qi(x)fi(x)−fi+1(x),degfi+1 (x)<degfi(x)
.
.
.
fs−1(x) = qs(x)fs(x),(fs+1(x) = 0)
In other words, perform the Euclidean algorithm and change the sign of the
remainder at each stage. It should be noted here that there exists a more general
definition of Sturm sequence, but the one given here suits our purpose.
Sturm’s Theorem. Let f(x)be a polynomial of positive degree in R[x]and let
f0(x) = f(x),f1(x) = f′(x), ..., fs(x)be the Sturm sequence for f(x). Assume
[a, b]is an interval such that f(a)6= 0 and f(b)6= 0. Then the number of distinct
real roots of f(x)in (a, b)is Va−Vb, where Vcdenotes the number of variations of
sign of {f0(c),f1(c), ..., fs(c)}.
Proofs can be found in [4] and [6]. We can get the total number of real roots by
looking at the limits as a→ −∞ and b→+∞. Thus, the total number of distinct
real roots will depend only on the leading terms of the polynomials in the Sturm
sequence.

REAL UNIVARIATE QUINTICS 3
Of course, in addition to Sturm’s Theorem, we will also use the theorem that
the complex roots of a real polynomial occur in conjugate pairs and the following
(which the reader can prove as an exercise, or see [1], p. 102 (Cor. 4.1.9) or [2], p.
17 (Exercise 8)), less well known
Theorem 1. If the discriminant of a real polynomial is not zero, then the sign
of the discriminant is (−1)r, where ris the number of complex conjugate pairs of
roots.
In the 1990’s, the authors Lu Yang, Songxian Liang, Jingzhong Zhang, Xiarong
Hou, and Zhenbing Zeng studied root multiplicities and obtained extensive results.
In [5], [8], and [9], those authors developed and applied the notions of complete
discrimination system, multiple factor sequence, and revised sign list, and proved
general theorems about conditions for root multiplicities.
The significance of this paper is 1) the conditions giving the order of the real
roots with respect to multiplicity are new and 2) the proof of the conditions for the
real and complex root multiplicities is the simplest and most transparent possible.
We use the computer software Maple to calculate the Sturm sequence for the
quintic. One obtains remainder polynomials of degrees 3, 2, 1, and 0, which we
suggestively name gcddeg3, gcddeg2, gcddeg1, and gcddeg0, respectively.
The Sturm sequence for the quintic is:
x5+px4+qx3+rx2+sx +t
5x4+ 4px3+ 3qx2+ 2rx +s
gcddeg3 = −2/5q−4
25 p2x3−3/5r−3
25 pqx2−4/5s−2
25 prx−t+ 1/25 ps
gcddeg2 = −
25
4−3p2q2+ 45 r2+ 8 r p3
−40 qs −38 rpq + 16 p2s+ 12 q3x2
2p2−5q2
−
25
2−25 tq + 4 q2r−3pr2
−21 spq −p2rq + 10 p2t+ 6 sp3+ 30 srx
2p2−5q2
−
25
4
−55 tpq + 16 tp3+ 75 tr −3psr + 4 q2s−p2sq
2p2−5q2
gcddeg1 = 8
25
(6000 p2trqs −7500 tq2sr −2490 sp2q2r2
−528 sp5q2r+ 1590 sp3q3r+ 2640 s2rp3q+ 588 sr2p4q−3300 q2s2rp
−1550 srpq4
−2400 p3tqr2
−584 p6tqr −3300 p3tq2s+ 1560 p4tq2r+ 1080 p5tq s −1200 p4trs −850 p2trq3
+ 56 p7sqr + 3000 tpq2r2+ 3250 tpq3s+ 4000 s3q2
−2200 s2q4+ 3125 t2q3
−1000 trq4+ 2925 sr2q3
+ 108 sp4q4+ 4185 s2p2q3
−2742 s2p4q2
−315 sp2q5
−24 sr2p6
−528 s2rp5
−3200 qs3p2
−460 p3r3q2
+ 4 p6r2q2
−36 p4r2q3+ 540 p2r4q+ 450 pr3q3+ 152 p5r3q+ 105 p2r2q4+ 216 p5tq 3
−630 p3tq4
−24 p7tq2+ 480 p5tr2+ 64 p8tr −112 p7ts −12 p6sq3+ 748 p6s2q+ 600 tpq 5
−3750 t2q2p2+ 1500 p4t2q
+ 640 p4s3+ 300 sq6
−108 p4r4
−16 p7r3
−100 r2q5
−675 r4q2
−200 p6t2
−72 p8s2)x
(−3p2q2+ 45 r2+ 8 rp3−40 qs −38 rpq + 16 p2s+ 12 q3)2

4 ELIAS GONZALEZ AND DAVID A. WEINBERG
−
4
25
(−5800 trp3qs + 7250 tq2sr p −1200 s2rq3+ 3800 tp2q2r2+ 1224 tp5q2r−11850 tp2q3s−3720 tp3q3r+ 7560 tp4q2s
−700 tr2p4q+ 1160 trp5s+ 8000 tqs2p2+ 3650 trpq 4
−332 p4s2rq + 1135 p2s2q2r−450 psr2q3+ 460 p3sq 2r2
−4p6sq2r+ 36 p4sq3r−540 p2sr3q−152 p5sr 2q−105 p2srq4
−2024 p6tsq −128 p7qrt −7500 p2t2qr
−900 tq6+ 64 p5s3+ 320 p7t2
−324 tp4q4+ 945 tp2q5
−4875 tr2q3
−16 tr2p6
−10000 tq2s2
+ 6500 tq4s−1600 tp4s2+ 108 p4sr 3+ 16 p7sr2+ 400 ps3q2+ 300 ps2q4
−315 p3s2q3+ 108 p5s2q2
+ 28 p6s2r−320 p3s3q+ 100 r q5s+ 675 r3q2s−6875 t2q3p+ 7500 t2q2p3+ 9375 t2q2r+ 36 p6q3t
−2700 p5t2q+ 1500 p4t2r+ 192 p8st −12 p7s2q)
(−3p2q2+ 45 r2+ 8 rp3−40 qs −38 rpq + 16 p2s+ 12 q3)2
gcddeg0 = 25
16
(−3p2q2+ 45 r2+ 8 rp3
−40 qs −38 rpq + 16 p2s+ 12 q3)2(256 p5t3
−192 p4qst2
−128 p4r2t2+ 144 p4rs2t+ 144 p3q2rt2
−6p3q2s2t−80 p3qr2st + 16 p3r4t−27 p2q4t2
+ 18 p2q3rst −4p2q2r3t−1600 p3qt3+ 160 p3rst2
−36 p3s3t+ 1020 p2q2st2+ 560 p2qr2t2
−746 p2qrs2t
+ 24 p2r3st −630 pq3rt 2+ 24 pq3s2t+ 356 pq2r2st −72 pqr 4t+ 108 q5t2
−72 q4rst + 16 q3r3t
+ 2000 p2rt3
−50 p2s2t2+ 2250 pq2t3
−2050 pqrst2+ 160 pqs3t−900 pr 3t2+ 1020 pr2s2t−900 q3st2
+ 825 q2r2t2+ 560 q2rs2t−630 q r3st −2500 pst3
−3750 qrt3+ 2000 qs2t2+ 108 r5t−27 r4s2+ 2250 r2st2
−1600 rs3t+ 256 s5+ 3125 t4
−27 p4s4+ 18 p3qrs3
−4p3r3s2
−4p2q3s3+p2q2r2s2
+ 144 p2qs4
−6p2r2s3
−80 pq2rs3+ 18 pq r3s2+ 16 q4s3
−4q3r2s2
−192 prs4
−128 q2s4+ 144 qr2s3)
(14 rp3qs −62 q2srp −132 ps2r−28 p3st −4r3p3
−4r2q3
−88 q2s2+ 12 q4s−18 p4s2
+ 125 t2q+p2q2r2
−3p2q3s+ 117 r2qs + 18 r3pq −6r2p2s+ 97 qs2p2
−40 tq2r
−300 trs −27 r4+ 160 s3
−66 p2qtr + 130 ptsq −6p3tq2+ 120 ptr2+ 24 pq 3t
−50 p2t2+ 16 p4tr)2(2 p2
−5q)2
Now let us do a Sturm analysis for this sequence.
The discriminant of the quintic is D= 256 p5t3−192 p4qst2−128 p4r2t2+
144 p4rs2t+ 144 p3q2rt2−6p3q2s2t−80 p3qr2st+ 16 p3r4t−27 p2q4t2+ 18 p2q3rst −
4p2q2r3t−1600 p3qt3+160 p3rst2−36 p3s3t+1020 p2q2st2+560 p2qr2t2−746 p2qrs2t+
24 p2r3st −630 pq3rt2+ 24 pq3s2t+ 356 pq2r2st −72 pqr4t+ 108 q5t2−72 q4rst +
16 q3r3t+ 2000 p2rt3−50 p2s2t2+ 2250 pq2t3−2050 pqrst2+ 160 pqs3t−900 pr3t2+
1020 pr2s2t−900 q3st2+ 825 q2r2t2+560 q2rs2t−630 qr3st −2500 pst3−3750 qrt3+
2000 qs2t2+ 108 r5t−27 r4s2+ 2250 r2st2−1600 rs3t+ 256 s5+ 3125 t4−27 p4s4+
18 p3qrs3−4p3r3s2−4p2q3s3+p2q2r2s2+ 144 p2qs4−6p2r2s3−80 pq2rs3+
18 pqr3s2+ 16 q4s3−4q3r2s2−192 prs4−128 q2s4+ 144 qr2s3.
Let L3, L2, and L1 denote the appropriate factors of the leading coefficients of
gcddeg3, gcddeg2, and gcddeg1, respectively:

REAL UNIVARIATE QUINTICS 5
L3 = 2p2−5q
L2 = 3 p2q2−45 r2−8rp3+ 40 qs + 38 rpq −16 p2s−12 q3
L1 = −54 r4+ 320 s3+ 250 qt2−176 q2s2+ 24 q4s−36 p4s2−100 p2t2
−124 srpq2+ 28 srp3q+ 260 sptq −132 p2qrt −12 p3tq2+ 48 ptq3
+ 2 p2q2r2−6q3sp2−80 q2rt + 194 qs2p2−600 str −264 ps2r+ 36 r3pq
−56 sp3t−12 sr2p2+ 234 sqr2+ 32 p4tr + 240 pr2t−8q3r2−8r3p3
Notice that the numerator of gcddeg0 is 25(L2)2D.
1.2. Real and Complex Root Multiplicities. The number of real roots is
determined by the leading coefficients of the Sturm sequence. However, in constructing
a proof, one has to be careful about whether or not some of these leading coefficients
are zero.
1.2.1. 5 Distinct Roots. The Sturm analysis shows that if L3>0, L2>0, L1>0,
and D > 0, then there are 5 distinct real roots. (Clearly, there is no other way
to get 5 real roots.) By Theorem 1, the condition for 3 real single roots and two
complex conjugate roots is D < 0. Again by Theorem 1, the conditions for one real
root and two distinct pairs of complex conjugate roots are D > 0 and (L3≤0 or
L2≤0 or L1≤0).
1.2.2. 1 Double and 3 Single Roots. We have D= 0 and L16= 0. There are 2 or 4
distinct real roots. First consider the case where L26= 0 and L36= 0. An immediate
and superficial consequence of Sturm’s Theorem is that to get 4 real roots, one must
have L3>0, L2>0, and L1>0. Otherwise there are 2 real roots. But there is an
interesting point here. A careful analysis of the Sturm sequence (as shown below)
shows that it suffices to consider L1 alone! Here is the Sturm analysis in this case.
x5x4(L3)x3(L2)x2(L1)x
−∞ − +−L3L2−L1
∞+ + L3L2L1

6 ELIAS GONZALEZ AND DAVID A. WEINBERG
L3L2L1
+ + + 4 −0 = 4
+ + −3−1 = 2
+−+ 2 −2 = 0 (Impossible)
+− − 3−1 = 2
−+ + 2 −2 = 0 (Impossible)
−+−1−3 = −2 (Impossible)
− − + 2 −2 = 0 (Impossible)
−−− 3−1 = 2
Thus, the Sturm analysis shows that there are 4 distinct real roots when L1>0,
and 2 distinct real roots when L1<0. In the special cases where L3 = 0 or L2 = 0,
it is impossible to get four sign changes in the Sturm analysis table, hence, there
must be only 2 distinct real roots. We would like to verify that the conditions in
these special cases still depend only on the original L1 (defined above).
Consider the special case where L2 = 0 and L36= 0. We know (by the superficial
consequence of Sturm’s Theorem) that there can only be 2 real roots. But we would
like to verify that this case forces L1<0. (Note that here we use f
L1, as it is not
the same coefficient of xas in the original Sturm sequence due to the special case
L2 = 0.)
x5x4(L3)x3(f
L1)x
−∞ − +−L3−f
L1
∞+ + L3f
L1
L3f
L1
+ + 2 −0 = 2
+−3−1 = 2
−+ 1 −1 = 0 (Impossible)
− − 2−2 = 0 (Impossible)
The Sturm analysis shows that L3 has to be greater than zero. If L2 = 0, then
s=−1/8−3p2q2+8 p3r−38 rpq+45 r2+12 q3
2p2−5qand substituting this into L1 gives
−1
64
(520 q3r+ 135 p3q3−18 p5q2+ 800 p2qt −252 pq4−1000 tq2−160 p4t
+ 1350 r3+ 604 p2q2r−380 p4qr −2205 pqr2+ 558 p3r2+ 48 p6r)2
(2 p2−5q)3.
(Notice the perfect square in the numerator, a result of computer algebra.) Therefore,
L1<0.

REAL UNIVARIATE QUINTICS 7
Now consider the special case where L3 = 0 and L26= 0. Let us recompute the
Sturm sequence for this case (solve L3 = 0 for qand substitute q= 2p2/5 into the
original quintic).
x5+px4+ 2/5p2x3+rx2+sx +t
5x4+ 4 px3+ 6/5p2x2+ 2 rx +s
new1gcddeg2 = −−6
125 p3+ 3/5rx2−−2
25 pr + 4/5sx+ 1/25 ps −t
new1gcddeg1 =
−50
27 −2700 srp5−82500 ps2r−187500 str+15000 sp3t+25500 sr2p2+75000 pr2t
−10500 p4tr+4200 p4s2+72 p8s+360 p7t−60 p6r2−16875 r4+100000 s3+2000 r3p3x
(2 p3−25 r)3−
25
27 −60 p6sr−16875 sr3+7625 p2s2r+51250 tp2r2+10500 p4st−6600 tp5r
+18750 p3t2−234375 rt2+2000 p3sr2−181250 psrt+216 tp8+250000 ts2−10000 ps3−450 p5s2
(2 p3−25 r)3
new1gcddeg 0 = −
27
500 (2 p3
−25 r)3(161250 p2s3r2+ 7031250 sr2t2
−5000000 s3rt −84375 s2r4
+ 337500 tr5+ 800000 s5+ 9765625 t4
−2160 p8str + 78000 p5r2st −202500 p4s2tr
−2062500 p3st2r−712500 p2str3+ 3187500 ps2r2t+ 1296 t2p10 + 480 p8s3
−75000 p5t3
+ 31625 p4s4
−54000 p7rt2+ 1800 p7ts2+ 90000 p6t2s+ 1200 p6r3t−300 p6r2s2
−17500 p5rs3+ 712500 p4r2t2
−40000 p3tr4+ 10000 p3s2r3+ 87500 p3ts3+ 1562500 p2t3r
+ 2343750 p2t2s2
−2812500 pt2r3
−600000 ps4r−7812500 pt3s−2700 srp5
−82500 ps2r−187500 str + 15000 sp3t+ 25500 sr2p2+ 75000 pr2t−10500 p4tr
+ 4200 p4s2+ 72 p8s+ 360 p7t−60 p6r2
−16875 r4+ 100000 s3+ 2000 r3p32
x5x4(f
L2)x2(f
L1)(f
L2)x
−∞ − +f
L2−(f
L1)(f
L2)
∞+ + f
L2 (f
L1)(f
L2)
f
L2f
L1 (f
L1)(f
L2)
+ + + 2 −0 = 2
+− − 1−1 = 0 (Impossible)
−+−3−1 = 2
− − + 2 −2 = 0 (Impossible)
f
L1 and f
L2 denote the appropriate factors of the leading coefficients in new1gcddeg1
and new1gcddeg2. It follows from the Sturm analysis that we must have f
L1>0.
Substitute q= 2p2/5 into L1: the result is L1 = −216
25 srp5−264 ps2r−600 str +
48 sp3t+408
5sr2p2+ 240 pr2t−168
5p4tr +32
5r3p3+336
25 p4s2+144
625 p8s+144
125 p7t−
24
125 p6r2−54 r4+ 320 s3. We see that f
L1 and L1 have opposite signs. This shows
that the condition is L1<0.
Finally consider the case L2 = 0 and L3 = 0. Recomputing the Sturm sequence and

8 ELIAS GONZALEZ AND DAVID A. WEINBERG
doing the Sturm analysis yields L1∗=p4−125s > 0, where L1∗is the appropriate
coefficient of the resulting gcddeg1. Furthermore, computer algebra shows that
L3 = 0 and L2 = 0 imply that the original L1 = −64(p4−125s)3
390625 . Thus, we must
have L1<0. Therefore, in all cases, the condition for one double root, one real
root, and two complex conjugate roots is L1<0.
1.2.3. One Triple and Two Single Roots or Two Double and One Single Root. We
have D= 0 and L1 = 0 and L26= 0. (By definition of Sturm sequence and
discriminant, it follows as a consequence that the constant term of gcddeg1 must
be zero.)
First consider the generic case L36= 0. The discriminant of gcddeg2 will tell whether
there is a triple root or 2 double roots, and in the latter case whether they are real
or complex conjugate. Here is the appropriate factor (having the same sign as) of
the discriminant of gcddeg2:
D2 = 160 s2q3+p4q2r2−236 p4qs2−136 p5st −3p4q3s+ 48 p5q2t−1100 tq3r−
128 p6rt −45 spr3+ 60 sq2r2−1380 p3r2t−8p2q3r2+ 337 p2q2s2+ 408 p3rs2−
12 p4r2s−357 p3q3t+ 6 p3qr3+ 24 p2q4s−500 p2qt2−24 pq2r3+ 660 pq4t+ 36 p6s2+
100 p4t2+ 9 p2r4−48 sq5+ 625 t2q2+ 1500 tq sr −4p5qrs + 1028 p4qr t + 11 p3q2rs +
800 p3qst−1735 p2q2rt+ 20 pq3rs+ 5475 pqr2t−600 sp2tr −1150 stpq2−1380 s2rpq−
3sp2r2q−3375 r3t+ 16 r2q4+ 900 r2s2.
If D2 = 0, then there is a triple root and 2 single roots.
If D2>0, then there are 2 double real roots and one single real root.
If D2<0, then there are 2 complex conjugate double roots and one single real root.
When there is a triple root, it is necessary to do a Sturm analysis.
x5x4(L3)x3(L2)x2
−∞ − +−L3L2
∞+ + L3L2
L3L2
+ + 3 −0 = 3
+−2−1 = 1
−+ 1 −2 = −1 (Impossible)
− − 2−1 = 1
Thus, if L2>0, then there are 3 real roots, and if L2<0, then there is 1 real
root.
Now consider the special case L3 = 0.

REAL UNIVARIATE QUINTICS 9
Then by the Sturm Theorem, there can only be 1 real root (one cannot get 3
sign changes).
(Only the triple root case needs the Sturm analysis because if there are 2 double
roots, then the sign of D2 tells whether they are real or complex.)
If L3 = 0, solve for qand recompute the Sturm sequence: q= 2p2/5. The result
is
x5+px4+ 2/5p2x3+rx2+sx +t
5x4+ 4 px3+ 6/5p2x2+ 2 rx +s
−−6
125 p3+ 3/5rx2−−2
25 pr + 4/5sx+ 1/25 ps −t
−50
27 15000 sp3t−2700 p5rs−187500 srt−82500 prs2−10500 p4rt+25500 p2r2s+75000 pr 2t
+2000 p3r3−60 p6r2+100000 s3+360 p7t+72 sp8−16875 r4+4200 s2p4x
(2 p3−25 r)3
−25
27
7625 p2s2r−16875 r3s+10500 p4st+51250 tp2r2−6600 tp5r+2000 p3sr2−60 p6sr−
181250 psrt−10000 ps3+216 tp8−234375 rt2+250000 ts2+18750 p3t2−450 p5s2
(2 p3−25 r)3
Let us denote by f
L2 and f
L1 the appropriate factors of the leading coefficients of
gcddeg2 and gcddeg1 from the new Sturm sequence. We need f
L26= 0 and f
L1 = 0.
If we plug in q= 2p2/5 into the appropriate factor of the numerator of the leading
coefficient of the original gcddeg2, then we get −9
125 2p3−25 r2. If we plug in
q= 2p2/5 into the appropriate factor of the numerator of the leading coefficient of
the original gcddeg1, then we get
−300 srt −132 prs2+204
5p2r2s+ 120 pr2t+ 24 sp3t−108
25 p5rs −84
5p4rt + 160 s3+
16
5p3r3+72
125 p7t+72
625 sp8+168
25 s2p4−12
125 p6r2−27 r4.
The Sturm analysis shows that if L2>0, there are 3 real roots, and if L2<0
there is 1 real root.
Notice that in the special case that L3 = 0, L2 is forced to be negative.
In the special case L3 = 0, it is also necessary to compare the discriminant of
the new gcddeg2 with that of the original gcddeg2. The discriminant of the new
gcddeg2 is
−24
3125 sp4+24
125 p3t+4
625 p2r2−4
125 prs +16
25 s2−12
5rt.
If we plug q= 2p2/5 into the appropriate factor (D2) of the original gcddeg2,
we get

10 ELIAS GONZALEZ AND DAVID A. WEINBERG
−36
52p3−25 r26sp4−150 p3t−5p2r2+ 25 prs + 1875 rt −500 s2.
Thus, D2 still distinguishes between the triple root and two double roots, even
in the special case L3 = 0.
1.2.4. 1 Quadruple and 1 Single or 1 Triple and 1 Double. We have D= 0, L1 = 0,
L2 = 0, and L36= 0. In this case, all roots are real.
In this case, we know that gcddeg3 must have a multiple root (either triple or
double). Here is the Sturm sequence for gcddeg3:
gcddeg3 = −2/5q−4
25 p2x3−3/5r−3
25 pqx2−4/5s−2
25 prx−t+ 1/25 ps
(gcddeg3)′=−32/5q−4
25 p2x2−23/5r−3
25 pqx−4/5s+2
25 pr
3gcddeg1 = −1
75 8p3r−80 sp2−3p2q2+ 10 prq + 200 sq −75 r2x
−5q+ 2 p2
−1
75
6p3s−150 p2t−p2rq −5qps + 5 pr2+ 375 qt −50 sr
−5q+ 2 p2
3gcddeg0 = −2
25 −5q+ 2 p2−5400 sp5t−108 p5srq + 2700 p4rqt + 45 p3srq2
+ 135000 p2srt −32000 s3p2+ 108 s2p6+ 67500 p4t2−337500 p2qt2
+ 33750 psq2t−16875 pr2qt −337500 srqt + 1275 p2sqr2+ 3375 p2rq2t
−1500 prs2q+ 80000 s3q−22500 s2r2+ 84375 r3t+ 421875 q2t2
+ 10 p3r3q−420 p4sr2+ 540 p4s2q+ 27 p4q3s−9p4r2q2−675 p3q3t
−13500 p3r2t+ 4200 p3rs2−2925 p2s2q2+ 1125 psr3+ 32 p5r3−225 p2r4,
8p3r−80 sp2−3p2q2+ 10 prq + 200 sq −75 r22
Therefore, by the results for the cubic [3], the conditions for a quadruple root are
D3 = 864
15625 sp5t+432
390625 p5srq −432
15625 p4rqt −36
78125 p3srq2−864
625 p2srt +1024
3125 s3p2−
432
390625 s2p6−432
625 p4t2+432
125 p2qt2−216
625 psq2t+108
625 pr2qt +432
125 srqt −204
15625 p2sqr2−
108
3125 p2rq2t+48
3125 prs2q−512
625 s3q+144
625 s2r2−108
125 r3t−108
25 q2t2−8
78125 p3r3q+
336
78125 p4sr2−432
78125 p4s2q−108
390625 p4q3s+36
390625 p4r2q2+108
15625 p3q3t+432
3125 p3r2t−
672
15625 p3rs2+468
15625 p2s2q2−36
3125 psr3−128
390625 p5r3+36
15625 p2r4= 0 (the discriminant
of gcddeg3), M1 = 8p3r−80 sp2−3p2q2+ 10 prq + 200 sq −75 r2= 0, where

REAL UNIVARIATE QUINTICS 11
M1 is the leading coefficient of 3gcddeg1 above. The condition for triple and double
root are D3 = 0 and M16= 0. (But it is unnecessary to say D3 = 0 because we
know gcddeg3 MUST have a multiple root.)
1.2.5. 1 Quintuple Root. The conditions are D= 0, L1 = 0, L2 = 0, and L3 = 0.
1.3. Order of the Real Roots with Respect to Multiplicity.
1.3.1. 1 Double and 3 Single Roots. We begin with the case where we have 1 double
real root and 3 single real roots. Solving gcddeg1 for the double root yields:
doubleroot =C0
C1
where C0= 48 sp4t+ 4 sp3r2+ 80 p3t2−32 p3rqt −3p3s2q+ 7 s2p2r−p2srq2−
4p2r2t+ 9 p2tq3−266 sqp2t+ 16 ps3+ 146 ptrq2−18 spr2q+ 290 sptr −275 pqt2+
12 ps2q2+ 4 sq3r−195 r2qt + 260 sq2t+ 27 sr3+ 375 t2r−36 q4t−48 rs2q−400 ts2
and C1= 2 (−28 sp3t−50 p2t2+ 120 pr2t−132 s2rp −6p3q2t+ 14 p3rsq −
62 prq2s−4p3r3−18 s2p4−88 s2q2+ 12 sq4−4r2q3+ 125 qt2+ 97 s2p2q−
6sp2r2−3p2q3s+p2q2r2+ 18 pr3q+ 117 sqr2−40 rq2t−300 srt −27 r4+
160 s3+ 130 qpst −66 p2rqt + 24 pq3t+ 16 p4tr)
come from the coefficients of gcddeg1.
We now translate the double root to the origin by substituting x=y+doubleroot
into the quintic. The resulting quintic then must have a factor of y2, as we have
shifted the double root to the origin. Here is the remaining cubic factor:
lef tovercubic =
y3+p+ 5 C0
C1y2+q+ 4 pC0
C1+ 10 C02
C12y+ 3 q C0
C1+r+ 6 pC02
C12+ 10 C03
C13
We now need information about how many real roots of a cubic x3+px2+qx+r
are positive. If we perform a Sturm sequence for the cubic, but this time checking
the variations in sign from 0 to ∞, we will find out how many of these single roots
are positive. The signs at 0 and ∞are determined by the constant terms and by the
leading coefficients. The cases actually condense, and depend only on the constant
terms in the Sturm sequence.
It actually takes several pages to write down the proof, using Sturm’s Theorem,
but the result is that if we are in the case of three single real roots, then if q > 0,
r > 0, and pq −9r > 0, then there are 0 positive single roots, and if q > 0, r < 0,
and pq −9r < 0, then there are 3 positive single roots. If neither of these cases
hold, then, if r > 0, then there are 2 positive single roots, and if r < 0, then there
is 1 positive single root. If we are in the case of one single real root, then if r > 0,

12 ELIAS GONZALEZ AND DAVID A. WEINBERG
then there are 0 positive single roots, and if r < 0, there is 1 positive single root.
By these results, when there are three single real roots, then if q+ 4 pC0
C1+ 10 C02
C12>
0, 3 q C0
C1+r+6 pC02
C12+10 C03
C13>0, and pC13q+4 p2C12C0−24 pC02C1−22 qC0C12−40 C03−9rC 13
C13>
0 , then there are 0 positive roots; if q+ 4 pC0
C1+ 10 C02
C12>0, 3 qC0
C1+r+ 6 pC02
C12+
10 C03
C13<0, and pC13q+4 p2C12C0−24 pC02C1−22 qC0C12−40 C03−9rC13
C13<0, then there
are 3 positive roots. If neither of these cases hold, then if 3 q C0
C1+r+6 pC02
C12+10 C03
C13>
0, then there are 2 positive roots, and if 3 qC0
C1+r+ 6 pC02
C12+ 10 C03
C13<0, then there
is 1 positive root.
1.3.2. 1 Double and 1 Single. doubleroot and lef tovercubic are the same as in the
preceding section. Then by the results on the number of positive or negative roots
for a cubic, if the constant term of lef tovercubic is positive, then the double root of
the quintic is bigger than the single root of the quintic, while if the constant term
of lef tovercubic is negative, then the single root of the quintic is bigger than the
double root of the quintic, i.e.
if 3 qC0
C1+r+ 6 pC02
C12+ 10 C03
C13>0, then double root >single root, while
if 3 qC0
C1+r+ 6 pC02
C12+ 10 C03
C13<0, then single root >double root.
But now we must consider the special cases where L2 = 0 or L3 = 0. In each of
these special cases, it is necessary to consider the recomputed Sturm sequences.
Computer algebra shows that in each of these special cases, the calculation of the
double root and the leftovercubic from the recomputed Sturm sequences coincide
with the result of substituting for s, respectively for q, (solving for s in L2 = 0,
respectively for qin L3 = 0), in doubleroot and leftovercubic from the original
Sturm sequence. Therefore, the conditions above for the relative position of the
double and single root hold in all cases.
1.3.3. 1 Triple and 2 Single. In this case the greatest common divisor of the quintic
and its derivative has degree 2. First consider the generic case where L36= 0. Recall
that the discriminant of gcddeg2 has been denoted D2. If D2 = 0, then there is
a triple root. This triple root is −(1/2) times the coefficient of xin the monic
associate of gcddeg2.
tripleroot =C2,1
L2
where C2,1= 6 sp3+ 4 q2r−3pr2−21 spq + 30 sr + 10 p2t−25 tq −p2rq
Now translate the triple root to the origin via y=x−tripleroot. The resulting
quintic will have a factor of y3. By computer algebra, the remaining quadratic
factor is

REAL UNIVARIATE QUINTICS 13
lef toverquadratic =y2+p+ 5 C2,1
L2 y+q+ 4 pC2,1
L2 + 10 C2,12
L2 2
The appropriate factor of the discriminant of this leftoverquadratic is
D22 = −4qL22−6pC2,1L2−15 C2,12+L22p2
If D22 <0, then the two single roots are complex. If D22 >0, then the two single
roots are real. If the constant term of lef toverquadratic is negative, then the
configuration is single <triple <single. If the constant term of lef toverquadratic
is positive, then if the coefficient of yin lef toverquadratic is negative, then triple
<single <single, while if the coefficient of yin lef toverquadratic is positive, then
single <single <triple.
Now consider the special case L3 = 0. It is necessary to consider the recomputed
Sturm sequence. But it follows from Sturm’s Theorem that it is impossible for the
two single roots to be real. It is interesting to note that computer algebra shows that
the discriminant of gcddeg2 in the recomputed Sturm sequence must be negative.
1.3.4. 2 Double and 1 Single. Again, in this case the greatest common divisor of the
quintic and its derivative must have degree 2. If D2<0, then the two double roots
are complex conjugate. If D2>0, then the two double roots are real. (gcddeg2)2
must divide the quintic. We can get the single root by solving for x in the quotient
of the quintic divided by (gcddeg2)2.
singleroot =C3
L2.
where C3=−34 p2rq + 8 rp4+ 44 spq −3p3q2−8sp3+ 12 pq3+ 57 pr2−16 q2r+
100 tq −120 sr −40 p2t
Then we can translate singleroot to the origin in gcddeg2 by substituting x=
y+singleroot into gcddeg2. The relative positions of the 2 double roots and 1
single root can then be determined from the coefficients of the resulting quadratic
polynomial. This quadratic polynomial is the following:
25
4(L2) y2+−25
2C2,1+25
2C3y+25
4
C2
3
(L2) +25
4C2,0−25
2
C3C2,1
(L2) 
where C2,1=−p2rq + 4 q2r−25 tq −21 spq + 30 sr + 10 p2t−3pr2+ 6 sp3
and C2,0=−16 tp3−75 tr + 3 psr −4q2s+ 55 tpq +p2sq
come from the coefficients of gcddeg2′slinear and constant terms. After making
this quadratic polynomial monic, we can say that if the resulting constant term is
negative, then we have double <single <double. If the resulting constant term
is positive, then if the resulting coeff of x is negative, we have single <double <
double, and if the resulting coeff of x is positive, we have double <double <single.
Furthermore note that only the case L3 NOT equal to 0 is relevant because if it is
zero, then by the Sturm analysis, you cannot have 3 real roots.

14 ELIAS GONZALEZ AND DAVID A. WEINBERG
1.3.5. 1 Quadruple and 1 Single. In this case, the greatest common divisor of the
quintic and its derivative has degree 3. A quadruple root of the quintic is a triple
root of the greatest common divisor. The quadruple root is thus −(1/3) times the
coefficient of x2in the monic associate of gcddeg3.
quadrupleroot = 1/25r−pq
−5q+2 p2.
We can then translate the quadruple root to the origin in the original quintic and
get a leftover linear factor whose root is the translated single root.
By computer algebra, we obtain that the translated single root is −1/2−15 pq+4 p3+25 r
−5q+2 p2.
Therefore, if −1/2−15 pq+4 p3+25 r
−5q+2 p2>0, then we have quadruple <single, while if
−1/2−15 pq+4 p3+25 r
−5q+2 p2<0, then we have single <quadruple.
1.3.6. 1 Triple and 1 Double. Again, the greatest common divisor of the quintic
and its derivative has degree 3. If this greatest common divisor does not have a
triple root, then it must have a double root (because the original quintic cannot
have 3 double roots), and the quintic must have 1 triple root and 1 double root.
We can now apply the results from our previous paper, [3], that determine the
conditions on the cubic that will determine whether double <single or single
<double, to gcddeg3, and this will give us the conditions on the quintic for
triple <double or double <triple. By computer algebra, the result is that if
27
4−1250 tq2+1000 tqp2−200 tp4+50 psq2−40 p3sq+8 p5s+375 r2q−150 r2p2
−150 rpq2+60 rp3q+15 p2q3−6p4q2−125 r3+75 r2pq−15 rp2q2+p3q3
(−5q+ 2 p2)3>0, then we have
double >triple, while if
27
4−1250 tq2+1000 tqp2−200 tp4+50 psq2−40 p3sq+8 p5s+375 r2q−150 r2p2
−150 rpq2+60 rp3q+15 p2q3−6p4q2−125 r3+75 r2pq−15 rp2q2+p3q3
(−5q+ 2 p2)3<0, then we
have triple >double.

REAL UNIVARIATE QUINTICS 15
2. Summary
Notation.
D= 256 p5t3−192 p4qst2−128 p4r2t2+ 144 p4rs2t+ 144 p3q2rt2−6p3q2s2t
−80 p3qr2st + 16 p3r4t−27 p2q4t2+ 18 p2q3rst −4p2q2r3t−1600 p3qt3
+ 160 p3rst2−36 p3s3t+ 1020 p2q2st2+ 560 p2qr2t2−746 p2qrs2t+ 24 p2r3st
−630 pq3rt2+ 24 pq3s2t+ 356 pq2r2st −72 pqr4t+ 108 q5t2−72 q4rst + 16 q3r3t
+ 2000 p2rt3−50 p2s2t2+ 2250 pq2t3−2050 pqrst2+ 160 pqs3t−900 pr3t2
+ 1020 pr2s2t−900 q3st2+ 825 q2r2t2+ 560 q2rs2t−630 qr3st −2500 pst3−3750 qrt3
+ 2000 qs2t2+ 108 r5t−27 r4s2+ 2250 r2st2−1600 rs3t+ 256 s5+ 3125 t4−27 p4s4
+ 18 p3qrs3−4p3r3s2−4p2q3s3+p2q2r2s2+ 144 p2qs4−6p2r2s3
−80 pq2rs3+ 18 pqr3s2+ 16 q4s3−4q3r2s2
−192 prs4−128 q2s4+ 144 qr2s3; the discriminant of the quintic
Let L3, L2, and L1 denote the appropriate factors of the leading coefficients of
gcddeg3, gcddeg2, and gcddeg1, respectively:
L3 = 2p2−5q; appropriate factor of the leading coefficient of gcddeg3
L2 = 40 qs −16 p2s−8rp3+ 38 rpq + 3 p2q2−12 q3−45 r2
L1 = −264 ps2r−12 p3tq2+ 36 r3pq −124 srpq2+ 28 srp3q+ 260 sptq −132 p2qrt
+ 240 pr2t+ 234 sqr2+ 32 p4tr + 48 ptq3−56 sp3t−80 q2rt + 194 qs2p2−600 str
−6q3sp2+ 2 p2q2r2−12 sr2p2−54 r4+ 320 s3−8q3r2−8r3p3+ 250 qt2
−176 q2s2+ 24 q4s−36 p4s2−100 p2t2
D2 = 24 p2q4s−1100 q3rt + 800 p3qst −1735 p2q2rt −3p2qr2s+ 20 pq3rs −600 p2rst
−1150 pq2st + 5475 pqr2t−1380 pqrs2+ 1500 qrst + 6 p3qr3+p4q2r2−128 p6rt
+ 660 pq4t−136 p5st −3p4q3s−236 p4qs2+ 337 p2q2s2+ 48 p5q2t−357 p3q3t
−12 p4r2s−45 pr3s+ 60 q2r2s−8p2q3r2−500 p2qt2−24 pq2r3−1380 p3r2t
+ 408 p3rs2−4p5qrs + 1028 p4qrt + 11 p3q2rs + 36 p6s2+ 100 p4t2+ 9 p2r4
−48 q5s+ 16 q4r2+ 160 q3s2+ 625 q2t2−3375 r3t+ 900 r2s2;
the discriminant of gcddeg2

16 ELIAS GONZALEZ AND DAVID A. WEINBERG
M1 = 8 p3r−80 sp2−3p2q2+ 10 prq + 200 sq −75 r2;
appropriate factor of the leading coefficient of 3gcddeg1
(from the Sturm sequence for gcddeg3)
C0= 48 sp4t+ 4 sp3r2+ 80 p3t2−32 p3rqt −3p3s2q+ 7 s2p2r−p2srq2
−4p2r2t+ 9 p2tq3−266 sqp2t+ 16 ps3+ 146 ptrq2−18 spr2q+ 290 sptr −275 pqt2
+ 12 ps2q2+ 4 sq3r−195 r2qt + 260 sq2t+ 27 sr3
+ 375 t2r−36 q4t−48 rs2q−400 ts2;
appropriate factor of the constant term of gcddeg1.
C1= 2 (−28 sp3t−50 p2t2+ 120 pr2t−132 s2rp −6p3q2t+ 14 p3rsq −62 prq2s
−4p3r3−18 s2p4−88 s2q2+ 12 sq4−4r2q3+ 125 qt2+ 97 s2p2q
−6sp2r2−3p2q3s+p2q2r2+ 18 pr3q+ 117 sqr2−40 rq2t−300 srt
−27 r4+ 160 s3+ 130 qpst −66 p2rqt + 24 pq3t+ 16 p4tr)
C2,1= 6 sp3+ 4 q2r−3pr2−21 spq + 30 sr + 10 p2t−25 tq −p2rq;
numerator of tripleroot
D22 = −4qL22−6pC2,1L2−15 C2,12+L22p2;
appropriate factor of the discriminant of leftoverquadratic
C3=−34 p2rq + 8 rp4+ 44 spq −3p3q2−8sp3+ 12 pq3+ 57 pr2−16 q2r+ 100 tq
−120 sr −40 p2t; numerator of singleroot
C2,0=−16 tp3−75 tr + 3 psr −4q2s+ 55 tpq +p2sq;
appropriate factor of the constant term of gcddeg2
C4=−1/2−15 pq + 4 p3+ 25 r
−5q+ 2 p2; quadruplero ot
C5=27
4−1250 tq2+1000 tqp2−200 tp4+50 psq2−40 p3sq+8 p5s+375 r2q−150 r2p2−150 rpq 2
+60 rp3q+15 p2q3−6p4q2−125 r3+75 r2pq−15 rp2q2+p3q3
(−5q+ 2 p2)3

REAL UNIVARIATE QUINTICS 17
F1=q+4pC0
C1
+ 10 C2
0
C2
1
F2=3qC0
C1
+r+pC2
0
C2
1
+10C3
0
C3
1
F3=pC3
1q+ 4p2C2
1C0−24pC2
0C1−22qC0C2
1−40C3
0−9rC3
1
C3
1
F4=C2
3
L22+C2,0
L2−22C3C2,1
L22
F5=−2C2,1
L2+2C3
L2
F6=q+4pC2,1
L2+ 10 C2
2,1
L22
F7=p+5C2,1
L2
Note: Some of the items above are exhibited as rational functions. In the table
below, these can be used to form polynomial conditions by replacing the rational
function by the product of its numerator and denominator.

18 ELIAS GONZALEZ AND DAVID A. WEINBERG
Real Root Configurations.
1. 5 distinct real roots L3>0L2>0 and L1>0 and D > 0
2. 3 distinct real roots and 2
complex conjugate roots
D < 0
3. 1 real root and 4 distinct
complex single roots
D > 0 and (L3≤0 or L2≤0 or
L1≤0)
4. 1 double real root and 3 single
real roots
D= 0 and L1>0
4.a. single <double <single <single F2>0 and (F1≤0 or F3≤0)
4.b. double <single <single <single F1>0 and F2<0 and F3<0
4.c. single <single <single <double F1>0 and F2>0 and F3>0
4.d. single <single <double <single F2<0 and (F1≤0 or F3≥0)
5. 1 double real root and 1
single real root and 2 complex
conjugate roots
D= 0 and L1<0
5.a. single <double F2>0
5.b. double <single F2<0
6. 2 real double roots and 1 real
single root
D= 0 and L1 = 0 and L26= 0 and
D2>0
6.a. single <double <double F4>0 and F5<0
6.b. double <single <double F4<0
6.c. double <double <single F4>0 and F5>0
7. 2 complex conjugate double
roots and 1 single real root
D= 0 and L1 = 0 and L26= 0 and
D2<0
8. 1 triple root and 2 single real
roots
D= 0 and L1 = 0 and L2>0 and
D2 = 0
8.a. triple <single <single F6>0 and F7<0
8.b. single <triple <single F6<0
8.c. single <single <triple F6>0 and F7>0
9. 1 triple root and 2 complex
conjugate roots
D= 0 and L1 = 0 and L2<0 and
D2 = 0

REAL UNIVARIATE QUINTICS 19
10. 1 quadruple root and 1 single
root
D= 0 and L1 = 0 and L2 = 0 and
L36= 0 and M1 = 0
10.a. quadruple <single C4>0
10.b. single <quadruple C4<0
11. 1 triple root and 1 double root D= 0 and L1 = 0 and L2 = 0 and
L36= 0 and M16= 0
11.a. triple <double C5>0
11.b. double <triple C5<0
12. 1 quintuple root D= 0 and L1 = 0 and L2 = 0 and
L3 = 0
Complex Root Multiplicities.
1. 5 distinct roots D6= 0
2. 1 double root and 3 single roots D= 0 and L16= 0
3. 2 double roots and 1 single root D= 0 and L1 = 0 and L26= 0 and
D26= 0
4. 1 triple root and 2 single roots D= 0 and L1 = 0 and L26= 0 and
D2 = 0
5. 1 quadruple root and 1 single
root
D= 0 and L1 = 0 and L2 = 0 and
M1 = 0
6. 1 triple root and 1 double root D= 0 and L1 = 0 and L2 = 0 and
M16= 0
7. 1 quintuple root D= 0 and L1 = 0 and L2 = 0 and
L3 = 0

20 ELIAS GONZALEZ AND DAVID A. WEINBERG
3. Bibliography
[1] Basu, Saugata, Pollock, Richard, and Roy, Marie-Francoise. Algorithms in Real
Algebraic Geometry. First Edition. Springer-Verlag, Berlin, Heidelberg, 2003.
[2] Benedetti, Riccardo and Risler, Jean-Jacques. Real Algebraic and Semi-algebraic
Sets. Hermann, Editeurs des Sciences et des Arts, Paris, 1990.
[3] Gonzalez, Eli and Weinberg, David A. “Root configurations of real univariate
cubics and quartics”. arXiv: 1511.07489v2, [math.AC], 8 Jan 2018.
[4] Jacobson, Nathan. Basic Algebra I. 2nd ed. W.H. Freeman and Company. 1985.
[5] Liang, S. and Zhang. J. “A complete discrimination system for polynomials with
complex coefficients and its automatic generation.” Science in China E Vol. 42,
No. 2, April 1999. p. 113-128.
[6] Sottile, Frank. Real Solutions to Equations from Geometry. American Mathematical
Society. 2011.
[7] Weinberg, David and Martin, Clyde. “A note on resultants.” Applied Mathematics
and Computation Vol. 24, 1987. p. 303-309.
[8] Yang, L., Hou, X.R., and Zeng, Z.B. “A complete discrimination system for
polynomials.” Science in China E Vol. 39, No. 6, Dec. 1996. p. 628-646.
[9] Yang, Lu. “Recent advances on determining the number of real roots of paramateric
polynomials.” Journal of Symbolic Computation Vol. 28, 1999. p. 225-242.
San Antonio, Texas
E-mail address:elias.gonzalez@nisd.net
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX
79409
E-mail address:david.weinberg@ttu.edu
URL:www.math.ttu.edu/~dweinber