Counter-examples to quantifier elimination
for fewnomial and exponential expressions
Andrei Gabrielov
Department of Mathematics, Purdue University
W. Lafayette, IN 47907-1395
e-mail: [email protected]
Received March 17, 1997
1. Introduction. The famous Tarski-Seidenberg [10,11] theorem asserts that any real
semialgebraic expression with quantifiers is equivalent to a real semialgebraic expression
without quantifiers, i.e., quantifier elimination is possible for semialgebraic expressions. A
well known example (Osgood [9]) shows that this result cannot be extended to expressions
with the exponential function, even if all the variables remain bounded: an expression
x, y, z, ∃u ∈ [0, 1], y = xu, z = x exp(u)
defines a germ Z of a subanalytic set at the origin in R3 such that any analytic function
(and even any formal power series) vanishing on Z should be identically zero.
Gabrielov [3,4] showed the possibility of the bounded quantifier simplification for
real semianalytic expressions i.e., equivalence of any such expression, with real analytic
functions from a given algebra closed under differentiation, to an expression with analytic
functions from the same algebra, with only existential (or only universal) quantifiers, as
long as all the variables under the quantifiers remain bounded. This can be reformulated
as the possibility of the quantifier simplification in the algebra of “restricted” analytic
functions, i.e., functions y = f (x), x ∈ Rn , y ∈ R, such that, after embedding of Rn in
RPn and R in RP1 , the graph of f is a subanalytic subset of RPn × RP1 .
Wilkie [12] proved that unbounded quantifier simplification is possible for real semialgebraic expressions with the exponential function.
Denef and van den Dries [1] found that quantifier elimination is possible in the algebra
of all restricted analytic functions if we allow an additional operation of “bounded division”
(the ratio of two functions f and g defined as f /g in the area where |f | < |g| and 0
elsewhere). Van den Dries, Macintyre and Marker [2] showed that the same result remains
valid if we add “unbounded” polynomial, exponential, and logarithmic functions to the
algebra of restricted analytic functions.
Gabrielov’s results [4,5,6] (see also Gabrielov and Vorobjov [7]) imply that bounded
quantifier simplification is possible for Pfaffian functions (i.e., analytic functions satisfying a
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triangular system of Pfaffian differential equations, Khovanskii [8]). Moreover, an effective
estimate for the complexity of the quantifier simplification can be obtained, in terms of
the complexity of the original expression and of the Pfaffian functions involved.
A key step in this estimate is an estimate for the complexity of the frontier and closure
of a semi-Pfaffian set. According to [6], “restricted” closure and frontier of a semi-Pfaffian
set X (i.e., the closure and frontier within the domain of definition of the Pfaffian functions
involved in the definition of X) are semi-Pfaffian, with complexity effectively estimated in
terms of the complexity of X.
In this note, we show that this is not true for the “unrestricted” frontier and closure.
Closely related to the above questions is the possibility of quantifier elimination and
simplification for fewnomial semialgebraic expressions. Fewnomials (Khovanskii [8]) are
polynomials that can be built up from the independent variables and constants using
a “simple” formula (see Definition 1 below) with a bounded number of the additions,
multiplications, and integer power operations. Domain of definition of a fewnomial is
defined as the complement of the zeroes of expressions under all integer power operations.
An example is a polynomial with a few nonzero monomials, of the arbitrarily high degree.
The complexity of a fewnomial is defined in terms of the number of operations in its
formula.
According to [6], restricted frontier and closure of a fewnomial semi-algebraic set X is
again a fewnomial semi-algebraic set, with complexity estimated in terms of the complexity
of X. Combined with results of [4,5,7] this implies the possibility of the bounded quantifier
simplification for fewnomial expressions.
We present here an example where the result of elimination of a single variable from
two fewnomial equations cannot be defined by a fewnomial expression without quantifiers,
i.e., a quantifier-free formula equivalent to a “simple” expression cannot be “simple.” This
means that quantifier elimination is impossible for fewnomial expressions.
Our example implies that unrestricted frontier and closure of a fewnomial semialgebraic set is not fewnomial.
Based on this example, we show that bounded quantifier elimination is impossible
for expressions with the exponential function, even if an additional operation of bounded
division is allowed.
The problem of the possibility of unbounded quantifier simplification for Pfaffian and
fewnomial expressions remains open.
2. Counter-examples for fewnomial expressions.
Definition 1.
(Khovanskii [6].) A polynomial in n variables x = (x1 , . . . , xn ) is called
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a fewnomial of complexity m if it can be constructed from the constants and independent
variables with no more than m of the following operations:
1. Addition, subtraction or multiplication.
2. Raising to an arbitrary natural power.
A point x is regular if, whenever the operation 2 is applied to a polynomial P , we have
P (x) 6= 0. The domain of definition of a fewnomial is the set of all regular points.
A fewnomial expression of complexity m is defined by no more than m of the operations
that include 1, 2, and the following additional operations:
3. Binary relations >, <, and =.
4. Logical operations ∧ and ∨.
5. Quantifiers ∃ and ∀ applied to some of the variables.
A fewnomial expression is restricted if, whenever the operation 2 is applied to a polynomial
P , a restriction
(*) c < |P | < C
is added, with positive constants c and C.
Remark. Division and negative powers can be added to the operations 1 and 2, to define
rational fewnomial expressions, but these rational expressions are equivalent to fewnomial
expressions without division.
For polynomials with real coefficients in the real domain, it follows from Khovanskii’s
theory of Pfaffian functions [8] that the number of isolated zeroes of a system of n equations
with fewnomials of complexity m is bounded by a certain explicit function M (n, m). This
implies a uniform bound on the number of isolated roots of any fewnomial expression with
existential quantifiers of bounded complexity.
Example 1.
equations:
For an integer k > 1, consider the following system of two fewnomial
tk − xt = 1,
(y − t)k − x(y − t) = 1.
(1)
In the real domain, the result of elimination of t from these equations is a semi-algebraic
set Sk ⊂ R2 defined by a fewnomial expression of complexity m = 10, with one existential
quantifier:
Sk = {(x, y) ∈ R2 , ∃t ∈ R, tk − xt = 1, (y − t)k − x(y − t) = 1}.
(2)
We claim that Sk cannot be defined by a fewnomial expression without quantifiers of
complexity independent of k.
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To show this, consider the set SC ⊂ C2 defined by the complexification of (2):
SC = {(x, y) ∈ C2 , ∃t ∈ C, tk − xt = 1,
(y − t)k − x(y − t) = 1},
(3)
i.e., the set of the complex values (x, y) for which the two equations in (1) have a common
complex root t. The first of these two equations defines t as a k-valued function of x, with
ramification points xk = kk (1 − k)1−k . It is easy to check that the monodromy group of
this function is the group of all permutation of k elements. After substitution t = t(x),
the second equation defines k(k + 1)/2-valued function y(x) with yij (x) = ti (x) + tj (x),
for 1 ≤ i ≤ j ≤ k. Here ti (x) are the k values of the function t(x). Monodromy group
action on the values yij has two invariant subsets: i = j, of the size k, corresponding to
a subset {(y/2)k − xy/2 = 1} ⊂ SC , and i < j, of the size k(k − 1)/2, corresponding to
an irreducible algebraic subset S 0 ⊂ SC . For a fixed real value of x, the set Sk contains at
most 3 points (x, y), corresponding to the sums of at most 3 real roots of tk − xt = 1. For
a generic real value of x, the set S 0 contains at least [k/2] real points (x, y), counting the
sums of pairs of complex conjugate roots of tk − xt = 1.
Any fewnomial expression without quantifiers defining Sk or any part of Sk containing
a segment V of a real curve belonging to S 0 , should contain a non-zero fewnomial P (x, y)
vanishing on V . In this case, P should vanish also at all points of the irreducible set S 0 .
Hence the set {P = 0, x = const}, for a generic real x, contains at least [k/2] isolated real
points. This implies that P cannot be a fewnomial of complexity independent of k.
Remark.
For small x, the segment V in the above arguments can be chosen so that
both t and y − t are close to 1. This implies that Sk cannot be defined by a fewnomial
expression without quantifiers of complexity independent of k, even if we add restrictions
1/2 < |t| < 2, 1/2 < |y − t| < 2 to (3).
Example 2.
Consider the cone
W = {(x, y, z, t) ∈ R4 , z > 0, tk − xtz k−1 = z k , (yz − t)k − x(yz − t)z k−1 = z k }
over the set of real roots of (1). The set W is defined by a fewnomial expression of
complexity m = 19, independent of k. The frontier ∂W = W̄ \ W of the set W is equal
to {z = t = 0, (x, y) ∈ Sk . Thus ∂W is nor fewnomial, i.e., it cannot be defined by a
fewnomial expression without quantifiers of complexity independent of k.
3. Counter-examples for exponential expressions.
Definition 2.
An exponential fewnomial of complexity m is a function defined by no
more than m operations that include 1, 2, and
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6. Exponentiation exp(.).
An exponential expression of complexity m is defined by no more than m operations 1–6.
An exponential expression is restricted if, in addition to the restriction (*), each time
the exponentiation operation exp(P ) appears, a restriction
(**)
−C < P < C
is added, with a positive constant C.
Rational exponential expressions can be defined with the following additional operation:
7. Division, with the domain of definition of the ratio restricted to non-zero values of
the denominator.
A rational exponential expression is restricted if, in addition to the restrictions (*) and
(**), each time the division operation P/Q appears, a restriction
(***) |P/Q| < C
is added, with a positive constant C.
Khovanskii’s theory [6] guarantees that the number of isolated real roots of an existential exponential expression (without universal quantifiers) of complexity m does not
exceed a certain explicit function of m.
Results of [4-7] imply that a restricted exponential expression with bounded quantifiers is equivalent to a restricted existential exponential expression, with complexity not
exceeding a certain explicit function of complexity of the original expression. This is true
also for restricted rational exponential expressions.
Example 3.
Introducing new variables u = log(t) and v = log(y − t), and replacing
an integer k by a real variable s in (2), we define a fewnomial exponential expression with
existential quantifiers:
S = {(x, y, s) ∈ R3 , ∃t ∈ R, ∃u ∈ R, ∃v ∈ R,
exp(u) = t, exp(v) = y − t, exp(sz) − xt = 1, exp(su) − x(y − t) = 1}.
For a positive integer k, we have S ∩ {s = k} = Sk . The same arguments as before show
that this set (and even its restricted part) cannot be defined by any exponential expression
with division (actually, by any expression with Pfaffian functions meromorphic in the whole
complex plane). As before, it is easy to define the set S as a “boundary at infinity” of a
subset defined by exponential equations. Our example shows that such boundary cannot
be defined by an exponential expression.
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