Research article

Objective penalty function method for nonlinear programming with inequality constraints

  • Received: 01 October 2024 Revised: 17 November 2024 Accepted: 20 November 2024 Published: 27 November 2024
  • MSC : 90C30, 65K05

  • This paper presents a novel nonsmooth objective penalty function for inequality constrained optimization problems. A modified flattened aggregate function, which is a smooth approximation of the max-value function, is discussed. Then, the smooth objective penalty function that contains the flattened aggregate function is proposed, and the exactness of the new function is studied. Based on this, an objective penalty function algorithm is proposed and its convergence is proven under mild conditions. Because of the flattened aggregate function, the gradient computation can usually be greatly reduced for problems with many constraints. Numerical experiments are included to illustrate the efficiency of the new algorithm through a series of numerical examples, especially for solving problems with many constraints.

    Citation: Zhuolin Yan, Xiaowei Jiang, Siyao Wang. Objective penalty function method for nonlinear programming with inequality constraints[J]. AIMS Mathematics, 2024, 9(12): 33572-33590. doi: 10.3934/math.20241602

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  • This paper presents a novel nonsmooth objective penalty function for inequality constrained optimization problems. A modified flattened aggregate function, which is a smooth approximation of the max-value function, is discussed. Then, the smooth objective penalty function that contains the flattened aggregate function is proposed, and the exactness of the new function is studied. Based on this, an objective penalty function algorithm is proposed and its convergence is proven under mild conditions. Because of the flattened aggregate function, the gradient computation can usually be greatly reduced for problems with many constraints. Numerical experiments are included to illustrate the efficiency of the new algorithm through a series of numerical examples, especially for solving problems with many constraints.



    Determining the irreducibility of a polynomial has been one of the most intensively studied problems in mathematics. Among many irreducibility criteria for polynomials in Z[x], a classical result of A. Cohn [1] states that if we express a prime p in the decimal representation as

    p=an10n+an110n1++a110+a0,

    then the polynomial f(x)=anxn+an1xn1++a1x+a0 is irreducible in Z[x]. This result was subsequently generalized to any base b by Brillhart et al. [2]. In 2002, Murty [3] gave another proof of this fact that was conceptually simpler than the one in [2].

    In the present work, we are interested in studying the result of A. Cohn in any imaginary quadratic field. Let K=Q(m) with a unique squarefree integer m1, be a quadratic field. We have seen that the quadratic field K is said to be real if m>0 and imaginary if m<0. The set of algebraic integers that lie in K is denoted by OK. Indeed,

    OK={a+bσma,bZ},

    where

    σm:={mif m1 (mod 4),1+m2if m1 (mod 4)

    [4]. Clearly, OQ(i)=Z[i], the ring of Gaussian integers, where i=1. It is well known that OK is an integral domain and K is its quotient field. Then the set of units in OK[x] is U(OK), the group of units in OK.

    In general, we know that a prime element in OK is an irreducible element and the converse holds if OK is a unique factorization domain. A nonzero polynomial p(x)OK[x] is said to be irreducible in OK[x] if p(x) is not a unit and if p(x)=f(x)g(x) in OK[x], then either f(x) or g(x) is a unit in OK. Polynomials that are not irreducible are called reducible. For β=a+bσmOK, we denote the norm of β by

    N(β)={a2mb2   if m1 (mod 4),a2+ab+b2(1m4)   if m1 (mod 4).

    Clearly, N(β)Z for all βOK. To determine whether αOK is an irreducible element, we often use the fact that if N(α)=±p, where p is a rational prime, then α is an irreducible element [4].

    For α,βOK with α0, we say that α divides β, denoted by αβ, if there exists δOK such that β=αδ. For α,β,γOK with γ0, we say that α is congruent to β modulo γ and we write αβ (mod γ), if γ(αβ). By a complete residue system modulo β in OK, abbreviated by CRS(β) [5], we mean a set of |N(β)| elements C={α1,α2,,α|N(β)|} in OK, which satisfies the following.

    (i) For each αOK, there exists αiC such that ααi (mod β).

    (ii) For all i,j{1,2,,|N(β)|} with ij, we have αiαj (mod β).

    We have seen from [6] that

    C={x+yix=0,1,,a2+b2d1 and y=0,1,,d1} (1.1)

    is a CRS(β), where β=a+biZ[i] and d=gcd(a,b). It is clear that

    C:={x+yix=0,1,,max{|a|,|b|}1 and y=0,1,,d1}C.

    In 2017, Singthongla et al. [7] established the result of A. Cohn in OK[x], where K is an imaginary quadratic field such that OK is a Euclidean domain, namely m=1,2,3,7, and 11 [4]. Regarding the complete residue system (1.1), they established irreducibility criteria for polynomials in Z[i][x] as the following results.

    Theorem A. [7] Let β{2±2i,1±3i,3±i} or β=a+biZ[i] be such that |β|2+2 and a1. For a Gaussian prime π, if

    π=αnβn+αn1βn1++α1β+α0=:f(β),

    with n1, Re(αn)1, and α0,α1,,αn1C satisfying Re(αn1)Im(αn)Re(αn)Im(αn1), then f(x) is irreducible in Z[i][x].

    In the proof of Theorem A in [7], the inequality

    |β|3+1+4M2, (1.2)

    where M=(max{a,|b|}1)2+(d1)2 is necessary. It can be verified that for β=a+biZ[i], if |β|<2+2 and a1, then β{3±i,2±2i,2±i,1±3i,1±2i,1±i,3,2,1}. It is clear that the Gaussian integers 2±2i, 1±3i, and 3±i satisfy (1.2), while 2±i,1±2i,1±i,3,2,1 do not. Consequently, we cannot apply Theorem A for these numbers. However, there is an irreducibility criterion for polynomials in Z[i][x] using β=3 in [7].

    Theorem B. [7] If π is a Gaussian prime such that

    π=αn3n+αn13n1++α13+α0,

    where n3, Re(αn)1, and α0,α1,,αn1C satisfying the conditions

    Re(αn1)Im(αn)Re(αn)Im(αn1),
    Re(αn2)Im(αn)Re(αn)Im(αn2),
    Re(αn2)Im(αn1)Re(αn1)Im(αn2),

    then the polynomial f(x)=αnxn+αn1xn1++α1x+α0 is irreducible in Z[i][x].

    In 2017, Tadee et al. [8] derived three explicit representations for a complete residue system in a general quadratic field K=Q(m). We are interested in the first one and only the case m1 (mod 4) because the complete residue system in another case, m1 (mod 4) is inapplicable for our work. The CRS(β) for m1 (mod 4) in [8] is the set

    C:={x+ymx=0,1,,|N(β)|d1 and y=0,1,,d1}, (1.3)

    where β=a+bm and d=gcd(a,b).

    Recently, Phetnun et al. [9] constructed a complete residue system in a general quadratic field K=Q(m) for the case m1 (mod 4), which is similar to that in (1.3). They then determined the so-called base-β(C) representation in OK and generalized Theorem A for any imaginary quadratic field by using such representation. These results are as the following.

    Theorem C. [9] Let K=Q(m) be a quadratic field with m1 (mod 4). If β=a+bσmOK{0} with d=gcd(a,b), then the set

    C={x+yσmx=0,1,,|N(β)|d1 and y=0,1,,d1} (1.4)

    is a CRS(β).

    From (1.3) and (1.4), we have shown in [9] for any m<0, that the set

    C:={x+yσmx=0,1,,max{|a|,|b|}1 and y=0,1,,d1}C. (1.5)

    Moreover, if d=1, then C={0,1,,max{|a|,|b|}1}, while b=0 implies C={x+yσmx,y=0,1,,|a|1}=C.

    Definition A. [9] Let K=Q(m) be an imaginary quadratic field. Let βOK{0} and C be a CRS(β). We say that ηOK{0} has a base-β(C) representation if

    η=αnβn+αn1βn1++α1β+α0, (1.6)

    where n1, αnOK{0}, and αiC (i=0,1,,n1). If αiC (i=0,1,,n1), then (1.6) is called a base-β(C) representation of η.

    Theorem D. [9] Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). Let β=a+bmOK be such that |β|2+1m and a1+1m. For an irreducible element π in OK with |π||β|, if

    π=αnβn+αn1βn1++α1β+α0=:f(β)

    is a base-β(C) representation with Re(αn)1 satisfying Re(αn1)Im(αn)Re(αn)Im(αn1), then f(x) is irreducible in OK[x].

    Theorem E. [9] Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). Let β=a+bσmOK be such that |β|2+(9m)/4, a1, and a+(b/2)1. For an irreducible element π in OK with |π|>(9m)/4(|β|1), if

    π=αnβn+αn1βn1++α1β+α0=:f(β)

    is a base-β(C) representation with Re(αn)1 satisfying Re(αn1)Im(αn)Re(αn)Im(αn1), then f(x) is irreducible in OK[x].

    In this work, we first establish further irreducibility criteria for polynomials in OK[x], where K=Q(m) is an imaginary quadratic field, which extend Theorem D and Theorem E. We observe that the result for the case m1 (mod 4) is a generalization of Theorem B. Furthermore, we provide elements of β that can be applied to the new criteria but not to the previous ones.

    In this section, we establish irreducibility criteria for polynomials in OK[x], where K is an imaginary quadratic field. To prove this, we first recall the essential lemmas in [7,10] as the following.

    Lemma 1. [10] Let K=Q(m) be an imaginary quadratic field. Then |β|1 for all βOK{0}.

    We note for an imaginary quadratic field K that |α|=1 for all αU(OK).

    Lemma 2. [7] Let f(x)=αnxn+αn1xn1++α1x+α0C[x] be such that n3 and |αi|M (0in2) for some real number M1. If f(x) satisfies the following:

    (i) Re(αn)1, Re(αn1)0, Im(αn1)0, Re(αn2)0, and Im(αn2)0,

    (ii) Re(αn1)Im(αn)Re(αn)Im(αn1),

    (iii) Re(αn2)Im(αn)Re(αn)Im(αn2), and

    (iv) Re(αn2)Im(αn1)Re(αn1)Im(αn2),

    then any complex zero α of f(x) satisfies |α|<M1/3+0.465572 if |argα|π/6; otherwise

    Re(α)<32(1+1+4M2).

    We note that the inequality |α|<M1/3+0.465572 appears in Lemma 2 follows from the proof of the lemma in [7] as follows: It was shown in [7] that

    0=|f(α)αn|>|α|3|α|2M|α|2(|α|1)=:h(|α|)|α|2(|α|1), (2.1)

    where h(x)=x3x2M. To obtain such inequality, the authors suppose to the contrary that |α|M1/3+0.465572. One can show that h(x) is increasing on (,0)(2/3,). Since M1/3+0.465572>2/3, it follows that

    h(|α|)h(M1/3+0.465572)=0.396716M2/30.280872138448M1/30.115841163475170752>0.396716M2/30.280873M1/30.115842=M1/3(0.396716M1/30.280873)0.1158420.000001, since M1>0,

    which contradicts to (2.1).

    Now, we proceed to our first main results. To obtain an irreducibility criterion for the case m1 (mod 4), we begin with the following lemma.

    Lemma 3. Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). Let β=a+bmOK be such that a>1 and

    M:=(max{a,|b|}1)2m(d1)2, (2.2)

    where d=gcd(a,b). Then M1.

    Proof. If b=0, then M=(a1)2m(a1)2=1m(a1)>1. Now, assume that b0 and we treat two separate cases.

    Case 1: |b|a. Then M=(|b|1)2m(d1)2(|b|1)2=|b|11.

    Case 2: |b|<a. Then M=(a1)2m(d1)2(a1)2=a11.

    From every case, we conclude that M1.

    By applying Lemmas 1–3, we have the following.

    Theorem 1. Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). Let β=a+bmOK be such that |β|M1/3+1.465572 and a1+(3/2)((1+1+4M)/2), where M is defined as in (2.2). For an irreducible element π in OK, if

    π=αnβn+αn1βn1++α1β+α0=:f(β)

    is a base-β(C) representation with n3 and Re(αn)1 satisfying conditions (ii)–(iv) of Lemma 2, then f(x) is irreducible in OK[x].

    Proof. Suppose to the contrary that f(x) is reducible in OK[x]. Then f(x)=g(x)h(x) with g(x) and h(x) in OK[x]U(OK). We first show that either degg(x)1 and |g(β)|=1 or degh(x)1 and |h(β)|=1. It follows from degf(x)3 that g(x) or h(x) is a positive degree polynomial. If either degg(x)=0 or degh(x)=0, we may assume that h(x)=αOK. Then degg(x)=degf(x) and f(x)=αg(x) so that π=αg(β). Since π is an irreducible element and αU(OK), we obtain g(β)U(OK) and thus, |g(β)|=1. Otherwise, both degg(x)1 and degh(x)1, we have that π=g(β)h(β). Using the irreducibility of π again, we deduce that either g(β) or h(β) is a unit and hence, either |g(β)|=1 or |h(β)|=1, as desired.

    We now assume without loss of generality that degg(x)1 and |g(β)|=1. We will show that this cannot happen. Note that M1 by Lemma 3. Moreover, since αiC for all i{0,1,,n1}, where C is defined as in (1.5), we have

    |αi||(max{a,|b|}1)+(d1)m|=(max{a,|b|}1)2m(d1)2=M

    for all i{0,1,,n1}. Since degg(x)1, g(x) can be expressed in the form

    g(x)=εi(xγi),

    where εOK is the leading coefficient of g(x) and the product is over the set of complex zeros of g(x). It follows from Lemma 2 that any complex zero γ of g(x) satisfies either

    |γ|<M1/3+0.465572 or Re(γ)<32(1+1+4M2).

    In the first case, it follows from |β|M1/3+1.465572 that

    |βγ||β||γ|>|β|(M1/3+0.465572)1.

    In the latter case, it follows from a1+(3/2)((1+1+4M)/2) that

    |βγ|Re(βγ)=Re(β)Re(γ)=aRe(γ)>a32(1+1+4M2)1.

    From both cases, by using Lemma 1, we obtain

    1=|g(β)|=|ε|i|βγi|i|βγi|>1,

    which is a contradiction. This completes the proof.

    By taking β=3 together with m=1 in Theorem 1, we obtain Theorem B. This shows that Theorem 1 is a generalization of Theorem B. We will show in the next section that if β=a+biZ[i]{0} with b=0, then β=3 is the only element that can be applied to Theorem 1.

    Next, we illustrate the use of Theorem 1 by the following example.

    Example 1. Let K=Q(5), β=3+5OK, and π=906959685. Then d=1 and so C={0,1,2}. Note that M=(31)2+5(11)2=2, |β|=14>M1/3+1.465572, a=3>1+(3/2)((1+1+4M)/2), and π is an irreducible element because N(π)=(9069)2+5(5968)2=260331881 is a rational prime. Now, we have

    π=(13+85)β5+2β4+2β3+β2+2β+1

    is its base-β(C) representation with n=5 and Re(αn)=13 satisfying conditions (ii)–(iv) of Lemma 2.

    By using Theorem 1, we obtain that

    f(x)=(13+85)x5+2x4+2x3+x2+2x+1

    is irreducible in OK[x].

    Note from Example 1 that we cannot apply Theorem D to conclude the irreducibility of the polynomial f(x) because |β|=|3+5|<2+6=2+1m. Moreover, we see that a=3<1+6=1+1m.

    For the case m1 (mod 4), we start with the following lemma.

    Lemma 4. Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). Let β=a+bσmOK be such that a+(b/2)1+(3/2)((1+1+4M)/2) and

    M:=(max{|a|,|b|}1)2+(max{|a|,|b|}1)(d1)+(d1)2(1m4), (2.3)

    where d=gcd(a,b). Then M1.

    Proof. If b=0, then a1+(3/2)((1+1+4M)/2)>1. It follows that

    M=(a1)2+(a1)2+(a1)2(1m4)>(a1)2=a11.

    If a=0, then b/21+(3/2)((1+1+4M)/2)>1. Thus, b>2 and so

    M=(b1)2+(b1)2+(b1)2(1m4)>(b1)2=b1>1.

    Now, assume that |a|1 and |b|1. If |a|=1 and |b|=1, then M=0, yielding a contradiction because a+(b/2)1+(3/2)((1+1+4M)/2). Then |a|>1 or |b|>1. It follows from d1 that

    M(21)2+(21)(d1)+(d1)2(1m4)(21)2=1.

    By applying Lemmas 1, 2 and 4, we obtain an irreducibility criterion for the case m1 (mod 4) as the following theorem.

    Theorem 2. Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). Let β=a+bσmOK be such that |β|M1/3+1.465572 and a+(b/2)1+(3/2)((1+1+4M)/2), where M is defined as in (2.3). For an irreducible element π in OK, if

    π=αnβn+αn1βn1++α1β+α0=:f(β)

    is a base-β(C) representation with n3 and Re(αn)1 satisfying conditions (ii)–(iv) of Lemma 2, then f(x) is irreducible in OK[x].

    Proof. Suppose to the contrary that f(x) is reducible in OK[x]. Then f(x)=g(x)h(x) with g(x) and h(x) in OK[x]U(OK). It can be proved similarly to the proof of Theorem 1 that either degg(x)1 and |g(β)|=1 or degh(x)1 and |h(β)|=1. We may assume without loss of generality that degg(x)1 and |g(β)|=1. We will show that this cannot happen. By Lemma 4, we have M1. For i{0,1,,n1}, since αiC, it follows from the definition of C in (1.5) that

    |αi||(max{|a|,|b|}1)+(d1)(1+m2)|=|((max{|a|,|b|}1)+d12)+(d12)m|=(max{|a|,|b|}1)2+(max{|a|,|b|}1)(d1)+(d1)2(1m4)=M.

    The remaining proof is again similar to that of Theorem 1 by using Lemmas 1, 2 and Re(β)=a+(b/2).

    We illustrate the use of Theorem 2 by the following example.

    Example 2. Let K=Q(3), β=4σ3, and π=359278σ3. Then d=1 and so C={0,1,2,3}. Note that M=(41)2+(41)(11)+(11)2=3, |β|=13>M1/3+1.465572, a+(b/2)=3.5>1+(3/2)((1+1+4M)/2), and π is an irreducible element because N(π)=3592359278+(278)2=106363 is a rational prime. Now, we have

    π=β4+3β3+β2+2β+1

    is its base-β(C) representation with n=4 and Re(αn)=1 satisfying conditions (ii)–(iv) of Lemma 2.

    By using Theorem 2, we obtain that

    f(x)=x4+3x3+x2+2x+1

    is irreducible in OK[x].

    From Example 2, we emphasize that we cannot apply Theorem E to conclude the irreducibility of the polynomial f(x) because |β|=|4σ3|<2+3=2+(9m)/4, although a=4>1 and a+(b/2)=4(1/2)>1.

    Let K=Q(m) be an imaginary quadratic field. In this section, we will try to find elements of β=a+bσmOK{0} that can be applied to Theorem 1, respectively, Theorem 2 but not to Theorem D, respectively, Theorem E. We are only interested in two cases, namely b=0 and b0 with d=gcd(a,b)=1 because the remaining case, b0 with d>1 requires us to solve a multi-variable system of inequalities, which is more complicated. To proceed with this objective, we begin with the following remarks.

    Remark 1. Let a and m be integers with m<0. Then the following statements hold.

    (i) a1+32(1+1+4(a1)2) if and only if a3.

    (ii) a1+32(1+1+41m(a1)2) if and only if a4+23+31m4.

    (iii) a1+32(1+1+4(9m)/4(a1)2) if and only if a4+23+3(9m)/44.

    Proof. For convenience, we let A=a1. We have for any real number x>0 that

    a1+32(1+1+4x(a1)2) ifandonlyif A34(1+1+4xA), ifandonlyif (43A31)21+4xA, ifandonlyif 16A23(83+12x)A30, ifandonlyif A[4A(23+3x)]0, ifandonlyif 4A(23+3x)0, ifandonlyif A23+3x4, ifandonlyif a4+23+3x4. (3.1)

    Substituting x=1, x=1m, and x=(9m)/4 in (3.1) lead to (i)–(iii), respectively, as desired.

    To compare Theorem 1 with Theorem D and to compare Theorem 2 with Theorem E, we require the following remark.

    Remark 2. For any real number x, the following statements hold.

    (i) 4+23+3x4(x+x)1/3+1.465572 for all x[3,).

    (ii) x2+5(x1)1/3+1.465572 for all x[1,).

    (iii) 3x+1(x1)1/3+1.465572 for all x[1,).

    (iv) x22+1(x1)1/3+1.465572 for all x[4,).

    (v) x(2(x1))1/3+1.465572 for all x[2.85,).

    (vi) x2+1(x1)1/3+1.465572 for all x[3,).

    (vii) 73121x>4+9x for all x(,2].

    (viii) 299x>4+9x for all x(,3].

    Proof of Remark 2. By using the WolframAlpha computational intelligence (www.wolframalpha.com), it can be verified by considering the graphs of both left and right functions of each inequality.

    Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). In this subsection, we will find elements of βOK{0} that can be applied to Theorem 1 but not to Theorem D. Now, let β=a+bm be a nonzero element in OK that can be applied to Theorem 1 but not to Theorem D. Then |β|M1/3+1.465572 and a1+(3/2)((1+1+4M)/2), where M is defined as in (2.2). Since β cannot be applied to Theorem D, one can consider two possible cases, namely, |β|<2+1m or |β|2+1m as follows:

    Case A: |β|<2+1m. Then, we now try to find elements of β that satisfy the following inequality system:

    |β|<2+1m|β|M1/3+1.465572a1+32(1+1+4M2). (3.2)

    We consider two cases as follows:

    Case 1: b=0. Then β=a and M=(a1)2m(a1)2=1m(a1). Thus, the system (3.2) becomes

    a<2+1m (3.3)
    a(1m(a1))1/3+1.465572 (3.4)
    a1+32(1+1+41m(a1)2). (3.5)

    By (3.5) and Remark 1(ii), we have a(4+23+31m)/4, which together with (3.3) yield

    4+23+31m4a<2+1m. (3.6)

    To show that the integers β=a satisfying (3.6) are solutions of the system above, we must show that they also satisfy (3.4). If m=1, then a(4+23+32)/42.93. It follows from Remark 2(v) with x=a that a(2(a1))1/3+1.465572=(1m(a1))1/3+1.465572. Assume that m2. By taking x=1m in Remark 2(i), we obtain that

    4+23+31m4(1m+1m)1/3+1.465572=(1m(2+1m1))1/3+1.465572>(1m(a1))1/3+1.465572, by (3.3),

    implying (3.4).

    We note for m=1 that the inequality (3.6) implies a=3. Hence, β=3Z[i] is the only element that can be applied to Theorem 1 but not to Theorem D.

    Case 2: b0 and d=1. There are two further subcases:

    Subcase 2.1: |b|a. Then |β|=a2mb2 and M=(|b|1)2=|b|1. Thus, the system (3.2) becomes

    a2mb2<2+1m (3.7)
    a2mb2(|b|1)1/3+1.465572a1+32(1+1+4(|b|1)2). (3.8)

    Since |b|a, we obtain from (3.8) that a1+(3/2)((1+1+4(a1))/2). Using Remark 1(i), we have that a3. It follows from |b|a, a3, and m1 that

    a2mb2a2ma2=a2(1m)9(1m)=31m>2+1m,

    which is contrary to (3.7). Thus, the system above has no integer solution (a,b). This means that the assumptions in the system generate no pairs (a,b) that are solutions to Theorem 1 and that are also not solutions to Theorem D.

    Subcase 2.2: |b|<a. Then |β|=a2mb2 and M=(a1)2=a1. Thus, the system (3.2) becomes

    a2mb2<2+1m (3.9)
    a2mb2(a1)1/3+1.465572 (3.10)
    a1+32(1+1+4(a1)2). (3.11)

    Using Remark 1(i) and (3.11), we have a3. Since m1, we obtain (65m)2=25m260m+36>16m252m+36=4(94m)(1m), yielding 65m>2(94m)(1m). It follows that

    (94m1m)2=105m2(94m)(1m)>4

    and so 94m1m>2. If |b|2, then a2mb294m>2+1m, which is contrary to (3.9). Thus, |b|=1. Using (3.9) and a3, we have 9ma2m<2+1m and so 9a2<5+41m, i.e., 3a<5+41m. We next show that the pairs (a,b) with

    3a<5+41m and b=±1 (3.12)

    also satisfy (3.10). Since |b|=1, a3, and Remark 2(vi) with x=a, we have

    a2mb2=a2ma2+1(a1)1/3+1.465572,

    yielding (3.10). Thus, we conclude that the pairs (a,b) satisfying (3.12) are solutions of the system above.

    Case B: |β|2+1m. Since we cannot apply the element β to Theorem D, we have a<1+1m. Now, we try again to find elements of β that satisfy the following inequality system:

    |β|2+1ma<1+1m|β|M1/3+1.465572a1+32(1+1+4M2). (3.13)

    We consider two cases as follows:

    Case 1: b=0. Then a<1+1m<2+1m|β|=a, which is a contradiction. Hence, the system (3.13) has no integer solution β=a. In other words, the assumptions in the system generate no pairs (a,b) that are solutions to Theorem 1 and that are also not solutions to Theorem D.

    Case 2: b0 and d=1. There are two further subcases:

    Subcase 2.1: |b|a. Then |β|=a2mb2 and M=(|b|1)2=|b|1. Thus, the system (3.13) becomes

    a2mb22+1m (3.14)
    a<1+1m (3.15)
    a2mb2(|b|1)1/3+1.465572 (3.16)
    a1+32(1+1+4(|b|1)2). (3.17)

    Since |b|a, we obtain from (3.17) that a1+(3/2)((1+1+4(a1))/2). It follows from Remark 1(i) that a3. Since d=1, we have |b|>a. By using (3.15) together with a3, we have 3a<1+1m, implying m5. It can be verified by using (3.17) that |b|((43(a1)3)2+27)/36. Now, we have that

    3a<1+1m and a<|b|(43(a1)3)2+2736. (3.18)

    To show that the pairs (a,b) satisfying (3.18) are solutions of the system, it remains to show that they also satisfy (3.14) and (3.16). Since |b|>a3 and m<0, we obtain

    a2mb2>a2ma2=a1m31m>2+1m,

    yielding (3.14). From Remark 2(ii) with x=|b|, we have

    a2mb2>5+b2(|b|1)1/3+1.465572,

    showing (3.16).

    Subcase 2.2: |b|<a. Then |β|=a2mb2 and M=(a1)2=a1. Thus, the system (3.13) becomes

    a2mb22+1m (3.19)
    a<1+1m (3.20)
    a2mb2(a1)1/3+1.465572 (3.21)
    a1+32(1+1+4(a1)2). (3.22)

    Again, using Remark 1(i) and (3.22), we obtain a3. By using (3.20) together with a3, we have 3a<1+1m, implying m5. Using (3.19), we can verify that |b|(5m+41ma2)/(m). Now, we have that

    3a<1+1m and 5m+41ma2m|b|<a. (3.23)

    To show that the pairs (a,b) satisfying (3.23) are solutions of the system, it remains to show that they also satisfy (3.21). It follows from b21, m5, and Remark 2(ii) with x=a that

    a2mb2a2+5(a1)1/3+1.465572,

    yielding (3.21).

    From every case, we conclude that elements of β=a+bmOK{0} with m1 (mod 4) that can be applied to Theorem 1 but not to Theorem D are shown in the following tables.

    We note from Subcase 2.2 in Table 1 that the number of a roughly grows as 241m. To see this, since 841m>1, we have

    5+41m<41m+841m+4=(241m+2)2

    and so 3a<5+41m<241m+2. This means that the number of such a is approximately 241m.

    Table 1.  Case A: |β|<2+1m.
    β=a+bm, d=gcd(a,b) Integer solutions (a,b)
    Case 1: b=0 4+23+31m4a<2+1m and b=0
    Case 2: b0 and d=1
        Subcase 2.1: |b|a none
        Subcase 2.2: |b|<a 3a<5+41m and b=±1

     | Show Table
    DownLoad: CSV

    We note from Table 2 that the complicated lower bound in Subcase 2.2 is actually very close to 1. Indeed, we show that

    5m+41ma2m<2.

    Since m1, it follows that

    (43m)216(1m)=(9m224m+16)16+16m=9m28m=m(9m8)>0,

    showing (43m)2>16(1m) and so 43m>41m. Using 3a<1+1m, we have that 2+m21m<a29. It follows that

    0<3+21mm=(5m+41m)+(2+m21m)m<(5m+41m)a2m(5m+41m)9m=4m+41mm<4m+(43m)m=4.

    This shows that (5m+41ma2)/(m)<4=2, as desired.

    Table 2.  Case B: |β|2+1m.
    β=a+bm, d=gcd(a,b) Integer solutions (a,b)
    Case 1: b=0 none
    Case 2: b0 and d=1
        Subcase 2.1: |b|a 3a<1+1m and a<|b|(43(a1)3)2+2736
        Subcase 2.2: |b|<a 3a<1+1m and 5m+41ma2m|b|<a

     | Show Table
    DownLoad: CSV

    Let K=Q(m) be an imaginary quadratic field with m1 (mod 4). In this subsection, we find elements of βOK{0} that can be applied to Theorem 2 but not to Theorem E. Now, let β=a+bσm be a nonzero element in OK that can be applied to Theorem 2 but not to Theorem E. Then |β|M1/3+1.465572 and a+(b/2)1+(3/2)((1+1+4M)/2), where M is defined as in (2.3). Since β cannot be applied to Theorem E, one can consider two possible cases, namely, |β|<2+(9m)/4 or |β|2+(9m)/4 as follows:

    Case A: |β|<2+(9m)/4. Then we will find elements of β that satisfy the inequality system:

    |β|<2+9m4|β|M1/3+1.465572a+b21+32(1+1+4M2). (3.24)

    We consider two cases as follows:

    Case 1: b=0. Then β=a and M=(a1)2+(a1)(a1)+(a1)2(1m)/4=(9m)/4(a1). Thus, the system (3.24) becomes

    a<2+9m4 (3.25)
    a(9m4(a1))1/3+1.465572 (3.26)
    a1+32(1+1+4(9m)/4(a1)2). (3.27)

    By (3.27) and Remark 1(iii), we have that a(4+23+3(9m)/4)/4, which together with (3.25) yield

    4+23+3(9m)/44a<2+9m4. (3.28)

    To show that the integers β=a satisfying (3.28) are solutions of the system above, we must show that they also satisfy (3.26). By taking x=(9m)/4 in Remark 2(i) and using (3.25), we obtain that

    4+23+3(9m)/44(9m4+9m4)1/3+1.465572=[9m4(2+9m41)]1/3+1.465572>(9m4(a1))1/3+1.465572. (3.29)

    It follows from (3.28) and (3.29) that a>((9m)/4(a1))1/3+1.465572, yielding (3.26).

    Case 2: b0 and d=1. There are two further subcases:

    Subcase 2.1: |b||a|. Then |β|=a2+ab+b2(1m)/4 and M=(|b|1)2=|b|1. Thus, the system (3.24) becomes

    a2+ab+b2(1m4)<2+9m4 (3.30)
    a2+ab+b2(1m4)(|b|1)1/3+1.465572a+b21+32(1+1+4(|b|1)2). (3.31)

    In this subcase, we now show that the system has no integer solution (a,b). If a<0, then it follows from (3.31) that b>0 and so (b/2)11+(3/2)((1+1+4(b1))/2). Then b2(11+3)b+(19+43)0, implying b11. It follows from a21, a>1(b/2), b11, and Remark 2(vii) with x=m that

    a2+ab+b2(1m4)>1+(1b2)b+b2(1m4)=b2(1m4)+b+1121121m+484=1273121m>12(4+9m)=2+9m4, (3.32)

    which is contrary to (3.30). Thus, a0. If a=0, then |b|=1 because d=1. This contradicts to (3.31), so a1. If |b|=1, then a=1 and so (3.31) is false. Thus, |b|2 and so |b|>a because d=1. It follows from (3.31) and |b|2 that a+(b/2)>2.4 and so |b|+(b/2)>2.4. This implies that b2 or b5. If b=2, then we obtain that 2=|b|>a(3/2)((1+1+4(21))/2)>1.4, which is a contradiction. If b=3, then we obtain that 3=|b|>a(3/2)((1+1+4(31))/2)(1/2)>1.2, which implies that a=2. It follows that

    a2+ab+b2(1m4)=22+23+32(1m4)=14(499m+499m)>14(8+364m)=2+9m4,

    which is contrary to (3.30). If b4, then

    a2+ab+b2(1m4)1+4+16(1m4)=12(94m+94m)>12(4+9m)=2+9m4,

    which is contrary to (3.30). If b5, then

    a52a+b21+32(1+1+4(|b|1)2)1+32(1+1+4(51)2)>3.22,

    showing a6. Since b5 and a6, it follows from b=|b|>a that

    a2+ab+b2(1m4)>a2b2+b2(1m4)=b2(1m41)+a225(1m41)+36=14(6925m+6925m)>14(8+364m)=2+9m4,

    which is contrary to (3.30).

    Thus, in this subcase, we conclude that the assumptions in the system generate no pairs (a,b) that are solutions to Theorem 2 and that are also not solutions to E.

    Subcase 2.2: |b|<|a|. Then |β|=a2+ab+b2(1m)/4 and M=(|a|1)2=|a|1. Thus, the system (3.24) becomes

    a2+ab+b2(1m4)<2+9m4 (3.33)
    a2+ab+b2(1m4)(|a|1)1/3+1.465572 (3.34)
    a+b21+32(1+1+4(|a|1)2). (3.35)

    If a<0, then it follows from a+(b/2)>1 that b>0. Since |a|>|b|=b and (3.35), we obtain (b/2)1a+(b/2)>1+(3/2)((1+1+4(b1))/2), implying b11. Now, we have that a2>1, a>1(b/2), and b11. It can be proved similarly to (3.32) that

    a2+ab+b2(1m4)>2+9m4,

    which is contrary to (3.33). Thus, a0. If a=0 or a=1, then 0<|b|<|a|1, which is impossible so that a2. If b=1, then it follows from (3.35) that a(1/2)1+(3/2)((1+1+4(a1))/2), implying a4. By taking x=a in Remark 2(iii), we have

    a2a+1m4=a(a1)+1m43a+1(a1)1/3+1.465572,

    yielding (3.34). It can be verified by (3.33) with b=1 that a<(89m+25+1)/2. This shows that

    4a<89m+25+12, when b=1. (3.36)

    If b=1, then it follows from (3.35) that a+(1/2)1+(3/2)((1+1+4(a1))/2), implying a2. By taking x=a in Remark 2(iii), we have that

    a2+a+1m4=a(a+1)+1m43a+1(a1)1/3+1.465572,

    yielding (3.34). It can be verified by (3.33) with b=1 that a<(89m+251)/2 and thus

    2a<89m+2512, when b=1. (3.37)

    We next show for b2 or b2 that the system above has no integer solution (a,b). If b2, then a=|a|>|b|=b2 and so a3. It follows that

    a2+ab+b2(1m4)32+32+22(1m4)=12(16m+16m)>12(4+9m)=2+9m4,

    which is contrary to (3.33). If b=2, then we obtain from (3.35) that a11+(3/2)((1+1+4(a1))/2), implying a4. Since d=1 and b=2, we have that a5. Hence,

    a22a+1m=a(a2)+1m5(3)+1m=12(16m+16m)>12(4+9m)=2+9m4,

    which is contrary to (3.33). If b3, then we have a(3/2)a+(b/2)1+(3/2)((1+1+4(a1))/2). This implies that a5. Since a>|b|=b, we obtain that ba1 and so aba2+a. It follows from b3, a5, aba2+a, and Remark 2(viii) with x=m that

    a2+ab+b2(1m4)a2a2+a+b2(1m4)9(1m4)+5=12299m>12(4+9m)=2+9m4,

    which is contrary to (3.33).

    Thus, in this subcase, we obtain that the pairs (a,b) with b0 and d=1 satisfying (3.36) or (3.37) are integer solutions of the system (3.24).

    Case B: |β|2+(9m)/4. Since a+(b/2)>1 and we cannot apply β to Theorem E, it follows that a<1. Thus, we have to find elements of β that satisfy the following inequality system:

    |β|2+9m4, a<1|β|M1/3+1.465572a+b21+32(1+1+4M2). (3.38)

    Note that M1 by Lemma 4. Then b/21+(3/2)((1+5)/2)>2.4 and so b5. If b<|a|, then a6 and so a+(b/2)<a+b<a+|a|=0, which is a contradiction. Thus, b|a|=a and so M=(b1)2=b1. Hence, the system (3.38) becomes

    a2+ab+b2(1m4)2+9m4, a<1 (3.39)
    a2+ab+b2(1m4)(b1)1/3+1.465572 (3.40)
    a+b21+32(1+1+4(b1)2). (3.41)

    Since b5 and d=1, we have a1. It follows by (3.41) that (b/2)11+(3/2)((1+1+4(b1))/2), implying b11. Note that ba, b11, and d=1 imply b>a. That is, b<a1. Now, we have that

    b11 and 1+32(1+1+4(b1)2)b2a1 (3.42)

    To show that the pairs (a,b) satisfying (3.42) are solutions of the system, it remains to show that they also satisfy (3.39) and (3.40). Since a21, a>1(b/2), and b11, we obtain by Remark 2(vii) with x=m that

    a2+ab+b2(1m4)>1+(1b2)b+b2(1m4)=b2(1m4)+b+1121(1m4)+12=1273121m>12(4+9m)=2+9m4,

    showing (3.39). It follows from a21, a>1(b/2), m3, and Remark 2(iv) with x=b that

    a2+ab+b2(1m4)>1+(1b2)b+b2>b22+1(b1)1/3+1.465572,

    yielding (3.40), as desired.

    From every case, we conclude that elements of β=a+bσmOK{0} with m1 (mod 4) that can be applied to Theorem 2 but not to Theorem E are shown in the following tables.

    We note from Subcase 2.2 in Table 3 that when b=1, the number of a roughly grows as 44(9m). Otherwise, b=1 implies that the number of a roughly grows as 44(9m)+1. To see these, one can see that

    89m+25<89m+2044(9m)+25=(244(9m)+5)2

    and so 89m+25<244(9m)+5. If b=1, then

    4a<89m+25+12<244(9m)+62=44(9m)+3,

    showing that the number of such a is approximately 44(9m). If b=1, we obtain

    2a<89m+2512<244(9m)+42=44(9m)+2,

    showing that the number of such a is approximately 44(9m)+1.

    Table 3.  Case A: |β|<2+9m4.
    β=a+bσm, d=gcd(a,b) Integer solutions (a,b)
    Case 1: b=0 4+23+3(9m)/44a<2+9m4 and b=0
    Case 2: b0 and d=1
        Subcase 2.1: |b||a| none
        Subcase 2.2: |b|<|a| 4a<89m+25+12, when b=1,
    2a<89m+2512, when b=1

     | Show Table
    DownLoad: CSV

    From Table 4, one can verify that if b|a| and d=1, then b11 and

    4.2b21+32(1+1+4(111)2)b21+32(1+1+4(b1)2)b2a1.

    This implies that the number of possible values of a is at most (b/2)4.2, the greatest integer less than or equal to (b/2)4.2.

    Table 4.  Case B: |β|2+9m4.
    β=a+bσm, d=gcd(a,b) Integer solutions (a,b)
    b<|a| none
    b|a| and d=1 b11 and 1+32(1+1+4(b1)2)b2a1

     | Show Table
    DownLoad: CSV

    Let K=Q(m) be an imaginary quadratic field with OK its ring of integers. In this paper, further irreducibility criteria for polynomials in OK[x] are established which extend the authors' earlier works (Theorems D and E). Moreover, elements of βOK that can be applied to the new criteria but not to the previous ones are also provided.

    This work was supported by the Science Achievement Scholarship of Thailand (SAST) and Department of Mathematics, Faculty of Science, Khon Kaen University, Fiscal Year 2022.

    All authors declare no conflicts of interest in this paper.



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