Krawtchouk polynomials and quadratic semi-regular sequences
Stavros Kousidis

TL;DR
This paper establishes bounds on the degree of regularity for quadratic polynomial systems using Krawtchouk polynomials, linking algebraic geometry with orthogonal polynomial theory.
Contribution
It introduces a novel approach to analyze the degree of regularity by interpreting the Hilbert series through Krawtchouk polynomials, providing new theoretical insights.
Findings
Derived bounds for the degree of regularity
Connected algebraic systems with orthogonal polynomial theory
Enhanced understanding of semi-regular polynomial systems
Abstract
We derive lower und upper bounds for the degree of regularity of an overdetermined, zero-dimensional and homogeneous quadratic semi-regular system of polynomial equations. The analysis is based on the interpretation of the associated Hilbert series as the truncation of the generating function of values of a certain family of orthogonal polynomials, the Krawtchouk polynomials.
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Krawtchouk polynomials and quadratic semi-regular sequences
Stavros Kousidis
Federal Office for Information SecurityGodesberger Allee 185–189BonnGermany53175
Abstract.
We derive lower und upper bounds on the degree of regularity of an overdetermined, zero-dimensional and homogeneous quadratic semi-regular system of polynomial equations. The analysis is based on the interpretation of the associated Hilbert series as the truncation of the generating function of values of a certain family of orthogonal polynomials, the Krawtchouk polynomials.
Groebner bases; Semi–regular sequences; Degree of regularity; Hilbert regularity; Orthogonal polynomials; Krawtchouk polynomials
1. Introduction
Semi-regular sequences model generic homogeneous systems of polynomial equations as a generalization of regular sequences to the overdetermined case. They were designed to be algebraically independent, i.e. to have as few algebraic relations between them as possible, in order to assess the complexity of Faugère’s Gröbner basis algorithm F5 (Faugère, 2002). The essential complexity parameter in that assessment is the degree of regularity, which is built in to the design of semi-regular sequences as a threshold up to which algebraic independence is maintained.
The degree of regularity of a semi-regular sequence essentially coincides with its Hilbert regularity, and can be computed by the power series expansion of a rational function and its truncation at the first non-positive coefficient. Asymptotic estimates of the degree of regularity via the analysis of this rational function by the saddle-point method of asymptotic analysis have been given by Bardet et al. in (Bardet, 2004; Bardet et al., 2004, 2005, 2003).
We follow a different approach to the degree of regularity in that we interpret the Hilbert series as the truncation of the generating function of values of a certain family of orthogonal polynomials, the Krawtchouk polynomials (Krawtchouk, 1929). This will enable us to give various descriptions of the degree of regularity based on information about the location of extreme roots of the Krawtchouk polynomials. In particular, we will derive lower and upper bounds on the degree of regularity without any further restrictions on the systems we consider. That is, for any overdetermined, zero-dimensional and homogeneous quadratic semi-regular system of polynomial equations with degree of regularity denoted by , we establish the lower bounds
[TABLE]
where is the unique positive real root of the quartic polynomial
[TABLE]
Furthermore, for such we prove the upper bounds
[TABLE]
where is a particular positive real root of the sextic polynomial
[TABLE]
While the lower bounds are valid for any , the existence of the upper bounds depend on the conditions , and along with , respectively, which we will explain in detail.
The article is organized as follows. In § 2 we give a short introduction to semi-regular sequences and Krawtchouk polynomials, and explain the connection between them. In § 3 we relate the degree of regularity to the smallest root of Krawtchouk polynomials and translate information about the location of the smallest root to the degree of regularity. This involves an exact description of the degree of regularity as an eigenvalue problem as well as the translation of bounds. Since the eigenvalue problem seems to be intractable we focus on lower and upper bounds for the smallest root of Krawtchouk polynomials that are known to the literature, and derive the above claims in § 4, § 5, § 6, § 7. We conclude in § 8 with concrete values and comparisons for illustration purposes.
2. Semi-regular sequences and Krawtchouk polynomials
Let be a system of polynomial equations where is a field. We assume the system to be zero–dimensional, overdetermined and homogeneous quadratic, that is the graded commutative algebra is finite–dimensional, and the degree of each is . We will adopt the usual notation for graded algebras and ideals, that is and for an ideal generated by homogeneous elements .
Now, according to Bardet (Bardet, 2004), Bardet et al. (Bardet et al., 2004, 2005), Diem (Diem, 2015) and Hodge et al. (Hodges et al., 2017) such a system of polynomial equations is defined to be a semi-regular sequence when the multiplication with any is injective in the graded algebra up to a certain degree. To be precise, is semi-regular if the multiplication map
[TABLE]
is injective for each and where is the degree of regularity of the graded ideal given by
[TABLE]
By (Bardet et al., 2005, Proposition 5 (i)) and (Hodges et al., 2017, Theorem 2.3 (d)) the polynomial system is semi-regular if and only if the Hilbert series of is
[TABLE]
Here, means truncation at the first non-positive coefficient. That is,
[TABLE]
As noted in (Bardet et al., 2005, Proposition 5 (iii)) the degree of regularity of a semi-regular sequence is the index of the first non-positive coefficient of , i.e.
[TABLE]
and consequently coincides with the Hilbert regularity of the graded algebra . The degree of regularity is of great interest in the field of polynomial systems solving, since for semi-regular sequences the complexity of Faugère’s F5 algorithm (Faugère, 2002) for the computation of a Gröbner basis can be bounded by (Bardet et al., 2005, Proposition 5 (iv))
[TABLE]
where is the exponent in the complexity of matrix multiplication. The expansion of the polynomial allows the computation of the regularity for concrete instances when and are fixed. In particular, its -th coefficient for is
[TABLE]
The alternating summation makes this explicit formula combinatorially unstable. That is, from this description it is virtually impossible to establish meaningful conditions on that imply .
An alternative approach to the coefficients is to understand the polynomial as being the ordinary generating function of values of binary Krawtchouk polynomials at certain integers (see (4)). To recall those polynomials, we follow Levenshtein’s exposition (Levenshtein, 1995, (2)) (see also (Krasikov and Litsyn, 2001)) and denote by
[TABLE]
the (general) Krawtchouk polynomial of degree for . From this one can deduce the ordinary generating function (Levenshtein, 1995, (43)):
[TABLE]
The Krawtchouk polynomials are discrete orthogonal polynomials associated to the binomial distribution via the orthogonality relation (Levenshtein, 1995, Corollary 2.3)
[TABLE]
that holds for any . Here, denotes the Kronecker symbol. They can be computed from the recurrence relation (Levenshtein, 1995, Corollary 3.3)
[TABLE]
For our purposes we will only consider the binary Krawtchouk polynomials, that is , and drop this parameter to simplify the notation. Then, the ordinary generating function (2) simplifies to
[TABLE]
Let us compute some binary Krawtchouk polynomials (Cf. Figure 1).
[TABLE]
For further illustration we evaluate the above computed polynomials at .
[TABLE]
It is still challenging to unfold the recurrence relation (3) in order to predict such that . However, relation (4) allows a description of the degree of regularity (1) via roots of binary Krawtchouk polynomials as we will explain in § 3.
3. Roots of Krawtchouk polynomials and the degree of regularity
We collect some properties of roots of orthogonal polynomials.
Theorem 3.1 (Cf. (Szegő, 1975, Theorem 3.3.1, Theorem 3.3.2)).
Let denote the roots of the binary Krawtchouk polynomial where . We have,
- (1)
the roots of are real, distinct and are located in the interior of the interval , i.e. without loss of generality they are ordered as . 2. (2)
the roots of and interlace, i.e. for and we have .
The interlacing property allows to relate the degree of regularity of semi-regular sequences to the roots of binary Krawtchouk polynomials. In fact, this is the essential observation of this article.
Lemma \thelem.
Let be an overdetermined, zero-dimensional and homogeneous quadratic semi-regular sequence. The degree of regularity of is given by
[TABLE]
where denotes the smallest root of for each .
Proof.
Because of the interlacing property from Theorem 3.1 we have the following strictly decreasing sequence of smallest roots of the polynomials .
[TABLE]
Hence, implies for all . Conversely assume for all and . Since and the roots are distinct, there must be an even number such that
[TABLE]
We choose a minimal such and note that since (see (5)) and . By the interlacing property each interval
[TABLE]
contains exactly one root of . Since is even, the number of those intervals is odd and since we have either that contradicts the initial assumption, i.e. for all , or we have which contradicts the minimality of . Therefore,
[TABLE]
and for we have
[TABLE]
In particular, . ∎
By § 3 it is clear, that any useful expression for the smallest roots of binary Krawtchouk polynomials yields a description of the degree of regularity of semi-regular sequences. Levenshtein (Levenshtein, 1995) proves an expression based on the maximization of a quadratic form that we recollect.
Theorem 3.2 (Cf. (Levenshtein, 1995, Theorem 6.1)).
Let denote the smallest root of for each . Then,
[TABLE]
This allows to describe the determination of the degree of regularity of a semi-regular sequence as an eigenvalue problem.
Lemma \thelem.
Let be as in § 3. The degree of regularity of is given by
[TABLE]
where denotes the largest eigenvalue of the real symmetric tridiagonal matrix with non-zero entries only on the super- und subdiagonal as follows
[TABLE]
with and .
Proof.
This is a reformulation of § 3 via Theorem 3.2 and standard linear algebra. That is,
[TABLE]
with being non-zero on the superdiagonal as follows
[TABLE]
with . We can replace by the symmetric matrix , where is given in the formulation of § 3 above, without changing the quadratic form and obtain
[TABLE]
where denotes the largest eigenvalue of . Consequently, by § 3
[TABLE]
The tridiagonal matrix of § 3 is a Golub-Kahan matrix (Golub and Kahan, 1965). It appears that no explicit formulæ for the eigenvalues of such a matrix are known. Some general results on the explicit computation of eigenvalues of tridiagonal matrices are given by Kouachi (Kouachi, 2006). Unfortunately those results do not apply to our matrix.
Instead of producing an exact expression for the degree of regularity of semi-regular sequences, our § 3 allows us to immediately translate lower and upper bounds for the smallest root of binary Krawtchouk polynomials into bounds for the degree of regularity.
Lemma \thelem.
Let be as in § 3 with degree of regularity . Then,
[TABLE]
where and are (not necessarily strict) lower and upper bounds, respectively, for the smallest root of the binary Krawtchouk polynomial for each . If the bounds and are indeed strict, then they are allowed to attain the threshold , i.e.
[TABLE]
Proof.
For the first part one has to realize that
[TABLE]
and
[TABLE]
The threshold assertions about strict bounds are obvious. ∎
The following (strict) lower bounds on the smallest root of Krawtchouk polynomials have been reported in the literature.
Lemma \thelem ((Krasikov and
Zarkh, 2009, Corollary 1), (Levenshtein, 1995, (125)), (Szegő, 1975, (6.32.6))).
Consider the smallest root of the binary Krawtchouk polynomial . Then, for Krasikov and Zarkh (Krasikov and Zarkh, 2009, Corollary 1) give
[TABLE]
Furthermore, for each Levenshtein (Levenshtein, 1995, (125)) in combination with an upper bound on the largest root of the Hermite polynomial described by Szegő (Szegő, 1975, 6.32.6) gives
[TABLE]
where are the real zeroes of the Airy’s function that is a solution of the ordinary differential equation (see (Szegő, 1975, §1.81)). Note that and (Szegő, 1975, (6.32.7)).
We also consider the following (strict) upper bounds on the smallest root of Krawtchouk polynomials.
Lemma \thelem ((Levenshtein, 1983, (6.25)), (Levenshtein, 1995, (124)), (Szegő, 1975, (6.2.14))).
Consider the smallest root of the binary Krawtchouk polynomial . Then, for each Levenshtein (Levenshtein, 1995, (124)) in combination with a lower bound on the largest root of the Hermite polynomial described by Szegő (Szegő, 1975, (6.2.14)) gives
[TABLE]
Furthermore, for Levenshtein (Levenshtein, 1983, (6.25)) (Cf. (Krasikov and Litsyn, 2001, (74))) gives
[TABLE]
Figure 2 illustrates the lower, and Figure 3 additionally illustrates the upper bounds in a family of binary Krawtchouk polynomials. We will treat each of those bounds seperately to derive the corresponding bounds on the degree of regularity.
Note that there are further bounds present in the literature (Area et al., 2015, 2013; Jooste and Jordaan, 2014) that apply to binary Krawtchouk polynomials. The results in (Jooste and Jordaan, 2014, Theorem 3.2) give the upper bound and hence no extra information. The bounds established in (Area et al., 2013, Corollary 5.2) coincide with (7). The bounds given in (Area et al., 2013, Theorem 5.1 and Corollary 5.1) and (Area et al., 2015, Theorem 1) will be subject to future research.
4. Lower bound on the regularity following Krasikov and Zarkh
Theorem 4.1.
Let be as in § 3. The smaller root of the polynomial yields a lower bound for the degree of regularity as follows
[TABLE]
Proof.
By § 3 and (6) from § 3 we have
[TABLE]
where the maximum is taken over . Hence we seek the largest integer such that
[TABLE]
Now, for the term
[TABLE]
is monotonically increasing, since its derivative (in ) is positive for any choice of , and hence by simple evaluation at and one concludes that it takes values in . That is, we can simplify our consideration, seeking the largest integer such that
[TABLE]
since any such is valid also for (10) and hence gives a lower bound for the degree of regularity of . That is, we can equivalently consider the inequality
[TABLE]
The polynomial has a positive discriminant , and hence real roots given by
[TABLE]
Moreover, since we can identify our integer
[TABLE]
5. Lower bound on the regularity following Levenshtein and Szegő
Recall the real zero of the Airy’s function described in § 3.
Theorem 5.1.
Let be as in § 3. The quartic polynomial
[TABLE]
has a unique positive real root , and the degree of regularity of is bounded from below by
[TABLE]
Furthermore, with
[TABLE]
we have
[TABLE]
where
[TABLE]
Proof.
By § 3 and (7) from § 3 we have
[TABLE]
where the maximum is taken over . Hence we seek the largest integer such that
[TABLE]
Since this is equivalent to consider
[TABLE]
We do a variable substitution
[TABLE]
and obtain the Laurent polynomial
[TABLE]
Note that we consider only and that we have the rational function
[TABLE]
So we are interested in the roots of the nominator which is the quartic polynomial given above
[TABLE]
Its discriminant is negative for any since
[TABLE]
Therefore has two complex conjugated roots and two real roots . Moreover, since the constant term of the polynomial is negative there is a unique positive real root (Cf. Figure 4). Undoing the variable substitution (11) yields the claimed lower bound for the degree of regularity of . A symbolic computation in SageMath gives the expression for and finishes the proof. ∎
Let us focus on the asymptotic growth of the lower bound given in Theorem 5.1. We adopt the usual notation of asymptotically equivalent functions, that is iff .
Corollary \thecor.
Assume grows subquadratic in , i.e. . Then, as , the lower bound of Theorem 5.1 behaves as
[TABLE]
Proof.
We borrow the notation from Theorem 5.1. For we have , , and . Hence,
[TABLE]
and
[TABLE]
We omit a deeper asymptotic analysis involving monotonicity considerations for reasons of brevity, but further summarize some interesting cases.
Corollary \thecor.
Let and be real constants. Then, as , the lower bound of Theorem 5.1 behaves as
[TABLE]
Remark \therem.
Note that § 5 carries similarities with the summary of Gröbner basis computation costs in (Bardet et al., 2003, §6), though the corresponding polynomial equations systems differ.
Remark \therem.
In the case when grows quadratically in , i.e. for some positive constant , or when grows superquadratic in , i.e. , the lower bound given in Theorem 5.1 tends to the value . Those two cases behave as expected. As the number of quadratic semi-regular (i.e. in this sense algebraically indepent) equations becomes large, the Macaulay matrix already contains all homogeneous entries of degree whose total number is .
6. Upper bound on the regularity following Levenshtein and Szegő
Theorem 6.1.
Let be as in § 3. If the discriminant of the polynomial is non-negative, then its smaller root yields a lower bound for the degree of regularity as follows
[TABLE]
Proof.
By § 3 and (8) from § 3 we have
[TABLE]
where the minimum is taken over . Hence we seek the smallest integer such that
[TABLE]
We square (12) and obtain the inequality
[TABLE]
The roots of the quadratic polynomial
[TABLE]
are
[TABLE]
They are real in the case of a non-negative discrimant, i.e.
[TABLE]
Recall that we are interested in the smallest integer that satisfies (12). Hence, under the non-negativity condition (13) we have
[TABLE]
Remark \therem.
In contrast to the lower bounds established in Theorem 4.1 and Theorem 5.1, which exist for any , the upper bound in Theorem 6.1 depends on a non-negative discriminant . This can be interpreted in terms of the family of Krawtchouk polynomials. Our Figure 3 actually illustrates the non-negative case. In the case of a negative discriminant, the set from § 3, with being the upper bound of Levenshtein and Szegő (see (8) in § 3), is empty. That is, for any family member this upper bound does not pass .
7. Upper bound on the regularity following Levenshtein
Theorem 7.1.
Let be as in § 3. The sextic polynomial
[TABLE]
has a global maximum at some . If , then has a a unique real root . If , then
[TABLE]
Proof.
By § 3 and (9) from § 3 we have
[TABLE]
where the minimum is taken over . Hence we seek the smallest integer such that
[TABLE]
We do a variable substitution and square (14) to obtain
[TABLE]
Therefore we are interested in the roots of the sextic equation
[TABLE]
The sextic has a local extremum at and by Rolle’s lemma local extrema inside and (Cf. Figure 5). We look at the derivative of , that is
[TABLE]
The discriminant of the quartic factor of the derivative is
[TABLE]
and hence negative for . Therefore has two complex conjugate roots and two real roots . This shows that the sextic has exactly three local extrema at , and . A second derivative test with some further computations show that has a local minimum at and local maxima at if . We now focus on the interval since the initial assumption and variable substitution puts the restriction on those that we consider valid to satisfy (15). Note that . That is, if , then by the intermediate value theorem we have a unique real root that satisfies (15). After undoing the variable substitution our satisfies (14) if is in the valid range, i.e. , and consequently . ∎
Remark \therem.
The sextic of Theorem 7.1 turns out to be irreducible with full Galois group for almost all combinations . Hence the methods of Hagedorn (Hagedorn, 2000) for solving a solvable sextic are not applicable. For almost all remaining combinations it factors into a linear and quintic polynomial with full Galois group . Again, methods for solving a solvable quintic (Dummit, 1991) do not apply. But in some of those cases the linear factor coincides with the root that gives our upper bound. For concrete instances though, the root can be determined by a numerical approximation via a root-finding algorithm.
Remark \therem.
The conditions for the existence of the upper bound in Theorem 7.1, i.e. and , can be interpreted in complete analogy to § 6.
Remark \therem.
The position of the local maximum of the sextic in Theorem 7.1 can be given explicitely by a symbolic computation in SageMath applied to the quartic factor in (16).
8. Concrete values and comparisons
The following is a collection of tables illustrating the lower bounds , from Theorem 4.1 and Theorem 5.1, and the upper bounds , from Theorem 6.1 and Theorem 7.1, respectively. They are put in contrast to the asymptotic estimates of Bardet et al. (Bardet et al., 2005, Theorem 1), where we simply omitted the asymptotic term. Note that the Airy function considered in (Bardet et al., 2005, (3)) which is a solution of the differential equation is not the Airy function considered here in (7) from § 3 and Theorem 5.1.
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
9. Acknowledgements
I would like to thank Max Gebhardt, Jernej Tonejc and Andreas Wiemers for helpful discussions.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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