+ il2^RIHt^RIHtRRltRRltR#))S)PolyNcVeRMRpV'dTp\V.VO5/VBp\V.VO5/VBpVP'dVP'g \R4hVPVP8Xg \R4hVPpVP 4^8gVP 4^8d^0#VP 4pV'gVP 4MTp\ 4pV^,EF=wrV^,EF)wrV P 4p V P 4p W8wdK.V P4pV P4pW, P'gKhV PW\^, ,4pV PW]^, ,4pVV, \W,4, pVP'gKV^8gVV9dKV ^8d+WPV4, P'gEKVPV4EK, EK@ V#)aiCompute the *dispersion set* of two polynomials. For two polynomials `f(x)` and `g(x)` with `\deg f > 0` and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as: .. math:: \operatorname{J}(f, g) & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\ & = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\} For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`. Examples ======== >>> from sympy import poly >>> from sympy.polys.dispersion import dispersion, dispersionset >>> from sympy.abc import x Dispersion set and dispersion of a simple polynomial: >>> fp = poly((x - 3)*(x + 3), x) >>> sorted(dispersionset(fp)) [0, 6] >>> dispersion(fp) 6 Note that the definition of the dispersion is not symmetric: >>> fp = poly(x**4 - 3*x**2 + 1, x) >>> gp = fp.shift(-3) >>> sorted(dispersionset(fp, gp)) [2, 3, 4] >>> dispersion(fp, gp) 4 >>> sorted(dispersionset(gp, fp)) [] >>> dispersion(gp, fp) -oo Computing the dispersion also works over field extensions: >>> from sympy import sqrt >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ') >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ') >>> sorted(dispersionset(fp, gp)) [2] >>> sorted(dispersionset(gp, fp)) [1, 4] We can even perform the computations for polynomials having symbolic coefficients: >>> from sympy.abc import a >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) >>> sorted(dispersionset(fp)) [0, 1] See Also ======== dispersion References ========== .. [1] [ManWright94]_ .. [2] [Koepf98]_ .. [3] [Abramov71]_ .. [4] [Man93]_ FTz!Polynomials need to be univariatez(Polynomials must have the same generator)r is_univariate ValueErrorgendegree factor_listsetLCis_zerocoeff_monomialr is_integershiftadd)pqgensargssamerfpfqJsunusedtmnanbnanm1bnm1alphas&&*, v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/dispersion.py dispersionsetr%sRM5tD  QA QA ???!///<== 55AEE>CDD %%C xxzA~as  B $"B AUU AIA A AvBBG$$$##CA#J/D##CA#J/DD[AadG+E###qyEQJ1ua''%.0999 EE%L+0 Hcn\W.VO5/VBpV'g\PpV#\V4pV#)aCompute the *dispersion* of polynomials. For two polynomials `f(x)` and `g(x)` with `\deg f > 0` and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as: .. math:: \operatorname{dis}(f, g) & := \max\{ J(f,g) \cup \{0\} \} \\ & = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \} and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`. Note that we make the definition `\max\{\} := -\infty`. Examples ======== >>> from sympy import poly >>> from sympy.polys.dispersion import dispersion, dispersionset >>> from sympy.abc import x Dispersion set and dispersion of a simple polynomial: >>> fp = poly((x - 3)*(x + 3), x) >>> sorted(dispersionset(fp)) [0, 6] >>> dispersion(fp) 6 Note that the definition of the dispersion is not symmetric: >>> fp = poly(x**4 - 3*x**2 + 1, x) >>> gp = fp.shift(-3) >>> sorted(dispersionset(fp, gp)) [2, 3, 4] >>> dispersion(fp, gp) 4 >>> sorted(dispersionset(gp, fp)) [] >>> dispersion(gp, fp) -oo The maximum of an empty set is defined to be `-\infty` as seen in this example. Computing the dispersion also works over field extensions: >>> from sympy import sqrt >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ') >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ') >>> sorted(dispersionset(fp, gp)) [2] >>> sorted(dispersionset(gp, fp)) [1, 4] We can even perform the computations for polynomials having symbolic coefficients: >>> from sympy.abc import a >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x) >>> sorted(dispersionset(fp)) [0, 1] See Also ======== dispersionset References ========== .. [1] [ManWright94]_ .. [2] [Koepf98]_ .. [3] [Abramov71]_ .. [4] [Man93]_ )r%rNegativeInfinitymax)rrrrrjs&&*, r$ dispersionr+s>X a*T*T*A   H F Hr&)N) sympy.corer sympy.polysrr%r+r&r$r/sz zR r&