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Early in high school algebra, quadratics chosen as examples by teachers and textbooks alike tend to have integer coefficients and to factorise over the integers. This can give the misleading impression that such quadratics are the norm. As students progress into calculus and begin regularly seeing quadratics that are not as ‘nice’, we hope they become disabused of this notion. Indeed, even if the coefficients of the quadratic are integers, the probability that the quadratic factorises over the integers tends to zero as the range from which the integers are drawn grows (see [1]). But what if we ask about a behaviour less restrictive than factorising, say merely having real roots? This is the problem that concerns this Article.

We present an optimal version of Descartes’ rule of signs to bound the number of positive real roots of a sparse system of polynomial equations in n variables with $$n+2$$ monomials. This sharp upper bound is given in terms of the sign variation of a sequence associated to the exponents and the coefficients of the system.

Let $f_n(z) = \sum_{k = 0}^n \varepsilon_k z^k$ be a random polynomial where $\varepsilon_0,\ldots,\varepsilon_n$ are i.i.d. random variables with $\mathbb{E} \varepsilon_1 = 0$ and $\mathbb{E} \varepsilon_1^2 = 1$. Letting $r_1, r_2,\ldots, r_k$ denote the real roots of $f_n$, we show that the point process defined by $\{|r_1| - 1,\ldots, |r_k| - 1 \}$ converges to a non-Poissonian limit on the scale of $n^{-1}$ as $n \to \infty$. Further, we show that for each $\delta > 0$, $f_n$ has a real root within $\Theta_{\delta}(1/n)$ of the unit circle with probability at least $1 - \delta$. This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form., Comment: 19 pages

How many irreducible polynomials have real roots which, when expressed as simple continued fractions, all have common tails? We show how to identify all such polynomials (they have degree at most s...

Following a line of inquiry regarding the exact number of real roots of a real polynomial, this investigation considers: Descartes’ Rule of Signs, the Budan–Fourier Theorem, and versions of...

Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real roots of f if and only if g is a sum of squares of rational polynomials modulo f. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial g is non-negative on the real roots of a non-zero polynomial f , when the above assumption is satisfied., Comment: Revista de la Union Matematica Argentina, 2022

We study the distribution of the discriminant D(P) of polynomials P from the class Pn(Q) of all integer polynomials of degree n and height at most Q. We evaluate the asymptotic number of polynomials P ∈ Pn(Q) having all real roots and satisfying the inequality |D(P)| ≤ X as Q→∞and X/Q2n−2→ 0.

Athanasiadis [‘A survey of subdivisions and local $h$-vectors’, in The Mathematical Legacy of Richard P. Stanley (American Mathematical Society, Providence, RI, 2017), 39–51] asked whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. We confirm this conjecture and prove that the local $h$-polynomials for all the Cartan–Killing types have only real roots. Our proofs use multiplier sequences and Chebyshev polynomials of the second kind.

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertok, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

Made available in DSpace on 2022-04-28T19:29:20Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-01-01 Well-rounded lattices have been a topic of recent studies with applications in wiretap channels and in cryptography. A lattice of full rank in Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. In this paper, we investigate the well-roundedness of lattices coming from polynomials with integer coefficients and real roots. Department of Mathematics São Paulo State University Department of Mathematics São Paulo State University

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova's theorem., some references were added and several proofs were elaborated

We are interested in the zero locus of a Chapoton’s F-triangle as a polynomial in two real variables x and y. An expectation is that (1) the F-triangle of rank l as a polynomial in x for each fixed y∈[0,1]has exactly l distinct real roots in [0,1], and (2) ith root x i (y) (1≤i≤l) as a function on y∈[0,1]is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of F-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above expectation is true. We formulate inductive conjectures and questions for further rank cases. This study gives a new insight on the zero loci of f +- and f-polynomials.

We study real sequences $\{a_{n}\}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i, Comment: 14 pages; improved bounds; section 5 and 6 on optimality added

I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than $$2c+1$$ , if d is odd, and t is greater or equal than $$\max (3,2c+1)$$ , if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms.

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of this random variable., 14 pages

We prove that if { a k } is a sequence of positive numbers, such that ( 2 j - 1 ) j + 1 2 j j a 2 j + 1 ? a 2 j ? 2 j - 1 2 j a 2 j - 1 for all j = 1 , 2 , ? , then for all n = 1 , 2 , ? , x ? 0 , π , ? k = 1 n a k sin ( k x ) ? 0 . An example is { a k } = { 1 , 1 2 , 1 2 , 3 4 2 , 1 3 , 5 6 3 , ? } = { 1 , 0.5 , 0.707 , 0.530 , 0.577 , 0.481 , ? } where a k = 1 / ( k + 1 ) / 2 for odd k , and ( k - 1 ) / ( k k / 2 ) , for even k . This improves the well-known Vietoris sine inequality, by relaxing the requirement that { a n } has to be a nonincreasing sequence.The proof is based on a Lukacs-type inequality and a result on positive trigonometric sums with "convex" coefficients (both established recently by the authors), the classical Sturm Theorem on the number of real roots of a polynomial, and a well-known comparison principle. The symbolic manipulation software MAPLE is used amply for various computations.

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f_n(t):= \frac{1}{\sqrt{n}}\sum_{k=1}^{n}a_k \cos(kt)+b_k\sin(kt), ~x\in [0,2\pi], \] where the sequences $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent copies of a stationary Gaussian process centered with variance one and correlation function $\rho$ with associated spectral measure $\mu_{\rho}$. We focus here on the case where $\mu_{\rho}$ is not purely singular and we denote by $\psi_{\rho}$ its density component with respect to the Lebesgue measure $\lambda$. Quite surprisingly, we show that the asymptotics of the number of real zeros $\mathcal{N}(f_n,[0,2\pi])$ of $f_n$ in $[0,2\pi]$ is not related to the decay of the correlation function $\rho$ but instead to the Lebesgue measure of the vanishing locus of $\psi_{\rho}$. Namely, assuming that $\psi_{\rho}$ is $\mathcal{C}^1$ with H\"older derivative on an open set of full measure, one establishes that \[ \lim_{n \to +\infty} \frac{\mathbb E\left[\mathcal{N}(f_n,[0,2\pi])\right]}{n}= \frac{\lambda(\{\psi_{\rho}=0\})}{\pi \sqrt{2}} + \frac{2\pi - \lambda(\{\psi_{\rho}=0\})}{\pi\sqrt{3}}. \] On the other hand, assuming a sole log-integrability condition on $\psi_{\rho}$, which implies that it is positive almost everywhere, we recover the asymptotics of the independent case, i.e. the limit is $\frac{2}{\sqrt{3}}$. Besides, with further assumptions of regularity and existence of negative moment for $\psi_{\rho}$, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence., Comment: 39 pages

Let a n , n ⩾ 1 be a sequence of independent standard normal random variables. Consider the random trigonometric polynomial T n (θ) = Σ =1 a j cos(jθ), 0 ⩽ θ ⩽ 2π and let N n be the number of real roots of T n (θ) in (0, 2π). In this paper it is proved that $\lim _{n \to \infty } \frac{{Var(N_n )}} {n} = c_0 $ where 0 < c 0 < ∞.

We extend previous work by Inkeri, Leeming and Delange on the number of real roots of the Bernoulli polynomials. By these earlier methods the number of real roots could not be determined exactly in many cases. We introduce a new method that enables us to determine precisely the number of real roots in virtually every case, with rare exceptions of approximately one in 1.5×108.

The following quadratic equation has no real roots: (1) (x+a)2+(x+b)2=0,a,b∈R,a≠b. If we expand (1) and use the fact that it’s discriminant is negative, we get 2ab

SummaryGiven a polynomial with all real roots, the Polynomial Root Dragging Theorem states that moving one or more roots of the polynomial to the right will cause every critical point to move to the right, or stay fixed. But what happens to the position of a critical point when roots are dragged in opposite directions? In this note we discuss the Polynomial Root Squeezing Theorem, which states that moving two roots, ri and rj, an equal distance toward each other without passing other roots, will cause each critical point to move toward (ri and rj)/2, or remain fixed.

The wave field and dispersion equations are found for a porous layer surrounded by two elastic half-spaces. The porous layer is described by the effective model of a medium in which elastic and fluid layers alternate. To investigate the normal waves, all real roots of dispersion equations are determined and their movements as the wave number increases are investigated. As a result, the dispersion curves of all normal waves are constructed and the dependence of normal waves on the parameters of the porous layer and elastic half-spaces is analyzed. Bibliography: 6 titles.

This article contains an investigation of the equitable basis for the Lie algebra \({\mathfrak{sl}_2}\). Denoting this basis by {x, y, z}, we have $$[x,y] = 2x + 2y, \quad [y,z] = 2y + 2z, \quad [z, x] = 2z + 2x.$$ We determine the group of automorphisms G generated by exp(ad x*), exp(ad y*), exp(ad z*), where {x*, y*, z*} is the basis for \({\mathfrak{sl}_2}\) dual to {x, y, z} with respect to the trace form (u, v) = tr(uv) and study the relationship of G to the isometries of the lattices \({L={\mathbb Z}x \oplus {\mathbb Z}y\oplus {\mathbb Z}z}\) and \({L^* ={\mathbb Z}x^*\oplus {\mathbb Z}y^*\oplus {\mathbb Z}z^*}\). The matrix of the trace form is a Cartan matrix of hyperbolic type, and we identify the equitable basis with a set of simple roots of the corresponding Kac–Moody Lie algebra \({\mathfrak{g}}\), so that L is the root lattice and \({\frac{1}{2} L^*}\) is the weight lattice of \({\mathfrak g}\). The orbit G(x) of x coincides with the set of real roots of \({\mathfrak g}\). We determine the isotropic roots of \({\mathfrak g}\) and show that each isotropic root has multiplicity 1. We describe the finite-dimensional \({\mathfrak{sl}_2}\)-modules from the point of view of the equitable basis. In the final section, we establish a connection between the Weyl group orbit of the fundamental weights of \({\mathfrak{g}}\) and Pythagorean triples.

We derive estimates in usual Sobolev norms for solutions of degenerate hyperbolic equations if degenerate coefficients admit only real roots. Loss of derivatives occurs due to the degeneracy.

We survey the most used bounds for positive roots of polynomials and discuss their efficiency. We obtain new inequalities on roots of polynomials. Then we give new inequalities on roots of orthogonal polynomials, obtained from the differential equations satisfied by these polynomials. Mathematics subject classification (2000): 12D10, 68W30.

Around 200 b.c. Apollonius of Perga wrote Conics, a collection of books that includes results on normals to curves. In 1545 Geronimo Cardano produced Ars magna, a major work that contains the solution of the cubic equation. In 1659 John Wallis published Tractatus duo, prior de cycloide, posterior de cissoide (Two Treatises, the First on the Cycloid, the Second on the Cissoid), which has the first rectification of an algebraic curve, a result that was achieved by his student William Neil. We will see that these three events, separated by centuries in time, are connected by a thread in the fabric of mathematics. Apollonius (ca. 262–190 b.c.) is best known for his extensive study of conic sections, Conics [1], [2], a collection of eight books. Heath [8, p. 158] says of this work that “Book V is of an entirely different order, indeed it is the most remarkable of the extant Books.” In Book V Apollonius investigates, for a given conic C and a point P not on C, the number of points Q on C for which the distance to P is a local maximum or minimum. In Propositions 27–30 of Book V, he shows that such an extreme line passing through P and Q is perpendicular to the tangent of C at Q. In modern terms, we say that these extreme lines are normals to the curve. This note is on the number of normals to the parabola y = x. We will see that the geometric problem of finding the number of normals to the parabola through a given point is equivalent to the algebraic problem of finding the

Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special attention is paid to the cube case., new title; several paragraphs reformulated; 23 pages

Nuij’s theorem states that if a polynomial p ∈ ℝ[z] is hyperbolic (i.e., has only real roots), then p+sp'' is also hyperbolic for any s ∈ ℝ. We study other perturbations of hyperbolic polynomials of the form pa(z, s) := . We give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which pa(z, s) is a pencil of hyperbolic polynomials. We also give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which the associated families pa(z, s) admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.

In this Note we give a proof of a generalized version of the classical Budan–Fourier theorem, interpreting sign variations in the derivatives in terms of virtual roots.

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial a 0 + a 1 x + a 2 x 2 + ⋯ + a n − 1 x n − 1 . a_0 +a_1 x+ a_2 x^2+ \cdots +a_{n-1}x^{n-1}. The coefficients a j a_j ( j = 0 , 1 , 2 , … , n − 1 ) (j=0, 1, 2, \dotsc , n-1) are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all n n sufficiently large, the above expected value is shown to be O ( log ⁡ n ) O(\log n) . Also, it is known that if the a j a_j have non-identical variance ( n − 1 j ) \binom {n-1}{j} , then the expected number of real zeros increases to O ( n ) O(\sqrt {n}) . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than O ( log ⁡ n ) O(\log n) . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain O ( log ⁡ n ) O(\log n) . In fact, so far the case of var ( a j ) = ( n − 1 j ) \mbox {var}(a_j)={\binom {n-1}{j}} is the only case that can significantly increase the expected number of real zeros.

Let a function f(x) ∈ C(r)[a,b], r ≥ 0, and take values that have different signs at the endpoints of the interval [a,b]. In the article, we propose effective methods for calculating real roots of the equation f(x)=0 on the given interval [a,b]. We substantiate this method and make a comparison between this method and the chord and Newton methods.

This paper provides an asymptotic estimate for the expected number of real zeros of algebraic polynomials $P_n (x)=a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1}x^{n-1} $, where $a_j$'s ($j=0,1,2,\cdots,n-1$) are a sequence of normal standard independent random variables with the symmetric property $a_j \equiv a_{n-1-j}$. It is shown that the expected number of real zeros in this case still remains asymptotic to $(2/\pi)\log n$. In the previous study, it was shown for the case of random trigonometric polynomials this expected number of real zeros is halved when we assume the above symmetric properties.

New algebraic criteria of absolute stability are obtained on the basis of a polynomial form of the frequency condition for absolute stability (Popov inequality). The analysis of the inequality is reduced to an analysis of the local properties of polynomials and a rational function on the negative real semiaxis. As a result, the stability criteria do not require the plotting of any sort of curves and can be expressed on the basis of algorithms of the Routh–Hurwitz type and calculations of the negative real roots of polynomials.

The article is concerned with the problem of estimating the stability margins for a system with a single negative feedback loop from the distribution of real roots of polynomials characterizing the frequency hodographs of the forward loop.

Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a double root. As an application, we consider the problem of estimating the number of real roots of $P_n$, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables $\xi$, that the expected number of real roots of $P_n(x)$ is exactly $\frac{2}{\pi} \log n +C +o(1)$, where $C$ is an absolute constant depending on the atom variable $\xi$. Prior to this paper, such a result was known only for the case when $\xi$ is Gaussian., Comment: 31 pages, 2 figures

We give a simple formula for the signature of a foldable triangulation of a lattice polygon in terms of its boundary. This yields lower bounds on the number of real roots of certain of systems of polynomial equations known as "Wronski systems"., Comment: 6 pages, 3 figures

Consider the n-th order (n ≥ 1 ) neutral differential equation where σ1 < σ 2 < ∞ and μ and η are increasing real-valued functions on [Ƭ1, Ƭ2] and [σ1, σ2] respectively. The function μ is assumed to be not constant on [Ƭ1, Ƭ2] and [Ƭ1, Ƭ2] for every Ƭ ∈ (Ƭ1, Ƭ2) similarly, for each σ ∈ (σ1, σ2), it is supposed that r\ is not constant on [σ1 , σ] and [σ, σ2]. Under some mild restrictions on Ƭ1,- and σ1, (ι = 1,2), it is proved that all solutions of (E) are oscillatory if and only if the characteristic equation of (E) has no real roots.

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains the list of all crystallographic arrangements, a large subclass of the class of simplicial hyperplane arrangements. Supposing that the rank is at least three, the classification yields Cartan schemes of type $A$ and $B$, an infinite family of series involving the types $C$ and $D$, and $74$ sporadic examples., 35 pages, 4 figures

We introduce the simple notion of a "crystallographic arrangement" and prove a one-to-one correspondence between these arrangements and the connected simply connected Cartan schemes for which the real roots are a finite root system (up to equivalence on both sides). We thus obtain a more accessible definition for this very large subclass of the class of simplicial arrangements for which a complete classification is known., 13 pages

Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$ possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$., 6 pages

The Mukhin-Tarasov-Varchenko Theorem, conjectured by B. and M. Shapiro, has a number of interesting consequences. Among them is a well-behaved correspondence between certain points on a Grassmannian - those sent by the Wronski map to polynomials with only real roots - and (dual equivalence classes of) Young tableaux. In this paper, we restrict this correspondence to the orthogonal Grassmannian OG(n,2n+1) inside Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and only if the corresponding tableau has a certain type of symmetry. From this we recover much of the theory of shifted tableaux for Schubert calculus on OG(n,2n+1), including a new, geometric proof of the Littlewood-Richardson rule for OG(n,2n+1)., Comment: 11 pages, color figures, identical to v1 but metadata corrected

We continue our study of Cartan schemes and their Weyl groupoids. The results in this paper provide an algorithm to determine connected simply connected Cartan schemes of rank three, where the real roots form a finite irreducible root system. The algorithm terminates: Up to equivalence there are exactly 55 such Cartan schemes, and the number of corresponding real roots varies between 6 and 37. We identify those Weyl groupoids which appear in the classification of Nichols algebras of diagonal type., 31 pages, 2 figures

A new short clear proof of the asymptotics for the number $c_n$ of real roots of the Bernoulli polynomials $B_n(x)$, as well as for the maximal root $y_n$: $$y_n=\frac{n}{2\pi e}+\frac{\ln(n)}{4\pi e}+O(1)\quad\text{and}\quad c_n=\frac{2n}{\pi e}+\frac{\ln(n)}{\pi e}+O(1).$$, Comment: 5 pages, 4 figures, rearranged figures

We present a simple algorithm for approximating all roots of a polynomial p(x) when it has only real roots. The algorithm is based on some interesting properties of the polynomials appearing in the Extended Euclidean Scheme for p(x) and p′(x). For example, it turns out that these polynomials are orthogonal; as a consequence, we are able to limit the precision required by our algorithm in intermediate steps. A parallel implementation of this algorithm yields a P-uniform NC2 circuit, and the bit complexity of its sequential implementation is within a polylog factor of the bit complexity of the best known algorithm for the problem.

SynopsisLetP(D)be a partial differential operator with constant coefficients. IfP(D)has a continuous linear right inverse in the space of tempered distributions, thenPis the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomialPis real and irreducible, thenP(D)admits a right inverse in the tempered distributions if and only ifP(×)has the property of zeros of R. Thorn.

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the A-discriminant, and give new bounds on the topology of certain A-discriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations, e.g., the number of smooth topological types attainable in certain families of real algebraic surfaces., Comment: 24 pages, 13 figures, final version with several improvements and small corrections