On multiplicative and additive properties of cyclotomic polynomials

Question

Is there explicit relation between $\Phi_{a+b}(x)$, $\Phi_{ab}(x)$, $\Phi_{a}(x)$ and $\Phi_{b}(x)$ at general coprime or non-coprime $a,b\in\Bbb Z$?

If $a,b$ are distinct primes then we have $x^{ab}-1=\Phi_{ab}(x)\Phi_{a}(x)\Phi_{b}(x)\Phi_{1}(x)$.

Answer 1

There isn't any explicit relation between $\Phi_{a}(x)$, $\Phi_{b}(x)$, and $\Phi_{a+b}(x)$ that I am aware of.

However between $\Phi_{ab}(x)$, $\Phi_{a}(x)$ and $\Phi_{b}(x)$, where $a$ and $b$ are primes I do know some basic observations:

The coefficients of $\Phi_{ab}(x)$ are in the range ${(-1, 0, 1)}$, and if $x^n$ is a term in $\Phi_{ab}(x)$ and $\gcd(ab,n)>1$, then the coefficient of $x^n$ is positive. That is, $-x^n$ is a term of $\Phi_{ab}(x)$, then $\gcd(ab,n)=1$. Another observation is that the second term and second to last terms have both negative coefficients. The last relations stil with $a$ and $b$ primes depend on weather $b = 1 \pmod a$. If this is true, then all terms of the form $x^{ad}$ up to the highest degree are included in $\Phi_{ab}(x)$ (and positive coefficients), however no terms of the form $x^{ad} are included. For example look at

$\Phi_{21}(x) = x^{12} - x^{11} + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1$

The terms in $\Phi_{21}(x)$ are $x^{12}, -x^{11}, x^9, -x^8, x^6, -x^4, x^3, -x, 1$. The degrees of these terms are $**12**, 11, **9**, 8, **6**, 4, **3**, 1, 0$. My last observation implies that all degrees that are multiples of $3$ (up to the highest degree) are included (bolded), but NO degrees with multiples of $7$ will be included.

In the case that $b 1 \pmod a$, then in $\Phi_{ab}(x)$ let $D(a) = {a, a_2, a_3,... a_n}$ denote all the degrees in $\Phi{ab}(x)$ which are multiples of $a$ and the same for $D(b) = {b, b_2, b_3,... b_n}$. Then both $D(a) and D(b)$ contain only consecutive multiples of $a$ and $b$. The same pattern is true for $\Phi_{33}(x)$ = x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1.

$\Phi_{35}(x)$ = x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1

$D(a) = 5, 10$ because $x^5$ and $x^10$ are the only terms which degrees are multiples of $5$. $D(b) = 7, 14$ because $x^7 and $x^14$ are the only terms which degrees are multiples of $7$. We see that $(5, 10)$ and $(7, 14)$ are consecutive multiples of $5$ and $7$.

$\Phi_{15}(x)$ = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1

$\Phi_{21}(x)$ = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1

$\Phi_{33}(x)$ = x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{35}(x)$ = x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1

$\Phi_{39}(x)$ = x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{51}(x)$ = x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^20 - x^19 + x^17 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{55}(x)$ = x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1

$\Phi_{57}(x)$ = x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{65}(x)$ = x^48 - x^47 + x^43 - x^42 + x^38 - x^37 + x^35 - x^34 + x^33 - x^32 + x^30 - x^29 + x^28 - x^27 + x^25 - x^24 + x^23 - x^21 + x^20 - x^19 + x^18 - x^16 + x^15 - x^14 + x^13 - x^11 + x^10 - x^6 + x^5 - x + 1

$\Phi_{69}(x)$ = x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{77}(x)$ = x^60 - x^59 + x^53 - x^52 + x^49 - x^48 + x^46 - x^45 + x^42 - x^41 + x^39 - x^37 + x^35 - x^34 + x^32 - x^30 + x^28 - x^26 + x^25 - x^23 + x^21 - x^19 + x^18 - x^15 + x^14 - x^12 + x^11 - x^8 + x^7 - x + 1

$\Phi_{85}(x)$ = x^64 - x^63 + x^59 - x^58 + x^54 - x^53 + x^49 - x^48 + x^47 - x^46 + x^44 - x^43 + x^42 - x^41 + x^39 - x^38 + x^37 - x^36 + x^34 - x^33 + x^32 - x^31 + x^30 - x^28 + x^27 - x^26 + x^25 - x^23 + x^22 - x^21 + x^20 - x^18 + x^17 - x^16 + x^15 - x^11 + x^10 - x^6 + x^5 - x + 1

$\Phi_{87}(x)$ = x^56 - x^55 + x^53 - x^52 + x^50 - x^49 + x^47 - x^46 + x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^27 - x^25 + x^24 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{91}(x)$ = x^72 - x^71 + x^65 - x^64 + x^59 - x^57 + x^52 - x^50 + x^46 - x^43 + x^39 - x^36 + x^33 - x^29 + x^26 - x^22 + x^20 - x^15 + x^13 - x^8 + x^7 - x + 1

$\Phi_{93}(x)$ = x^60 - x^59 + x^57 - x^56 + x^54 - x^53 + x^51 - x^50 + x^48 - x^47 + x^45 - x^44 + x^42 - x^41 + x^39 - x^38 + x^36 - x^35 + x^33 - x^32 + x^30 - x^28 + x^27 - x^25 + x^24 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1

$\Phi_{95}(x)$ = x^72 - x^71 + x^67 - x^66 + x^62 - x^61 + x^57 - x^56 + x^53 - x^51 + x^48 - x^46 + x^43 - x^41 + x^38 - x^36 + x^34 - x^31 + x^29 - x^26 + x^24 - x^21 + x^19 - x^16 + x^15 - x^11 + x^10 - x^6 + x^5 - x + 1

Notice that if the term $x^n$ in $\Phi_{ab}(x)$ is bold, then $\gcd(ab,n)>1$ and the coefficient is positive.