Edl.ddlmZddlmZddZddZdS))S)PolyNc|dnd}|r|}t|g|Ri|}t|g|Ri|}|jr|jstd|j|jkstd|j}|dks|dkrdhS|}|s|n|}t }|dD]\} } |dD]\} } | } | } | | kr5| }| }||z jsh| || dz z}| || dz z}||z t| |zz }|j s|dks||vr| dkr| | |z js| ||S)a=Compute 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]_ NFTz!Polynomials need to be univariatez(Polynomials must have the same generatorr)r is_univariate ValueErrorgendegree factor_listsetLCis_zerocoeff_monomialr is_integershiftadd)pqgensargssamer fpfqJsunusedtmnanbnanm1bnm1alphas 6lib/python3.11/site-packages/sympy/polys/dispersion.py dispersionsetr&sER +55tD   QA QA ?>!/><=== 5AE>ECDDD %C xxzzA~as  B $ ,"B AU 6A  IAv A AAv BBG$ ##C!A#J//D##C!A#J//DD[AadGG+E# qy EQJ 1u a!''%..09  EE%LLLL+ . Hcdt||g|Ri|}|s tj}nt|}|S)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,sFX a*T***T**A   FF Hr')N) sympy.corer sympy.polysrr&r,r'r%r0sfz z z z zR R R R R R r'